Answers
Answer:
P = 36 units
Step-by-step explanation:
Pythagoras
ip = √(12² + 9²)
= √(144 + 81)
= √225
= √15²
= 15 units
P = 9units + 12units + 15units = 36 units
Related Questions
What is a positive integer? what is a negative integer?.
Answers
Example: -1,-2,-3 zero is defined as nervier negative nor positive integer. Because zero is a whole number.
negative integer is any whole number that does have a - in front of it (eg. -1,-2,-3,-4)
Work out the coordinates that point P maps go after a single reflection in
a) the x-axis
b) the y-axis
Answers
The coordinates of point P after each reflection are given as follows:
a) x-axis: (3, -4).
b) y-axis: (-3, 4).
How to obtain the coordinates of point P?The original coordinates of point P are given as follows:
(3,4).
After a reflection over the x-axis, we change the sign of the y-coordinate, hence:
(3, -4).
After a reflection over the y-axis, we change the sign of the x-coordinate, hence:
(-3, 4).
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Consider the following sample of observations on coating thickness for low-viscosity paint:
.83 .88 .88 1.04 1.09 1.12 1.29 1.31 1.48 1.49 1.59 1.62 1.65 1.71 1.76 1.83
Assume that the distribution of coating thickness is normal (a normal probability plot strongly supports this assumption).
a. Calculate a point estimate of the mean value of coating thickness, and state which estimator you used.
b. Calculate a point estimate of the median of the coating thickness distribution, and state which estimator you used. c. Calculate a point estimate of the value that separates the largest 10% of all values in the thickness distribution from the remaining 90%, and state which estimator you used. [Hint: Express what you are trying to estimate in terms of m and s.]
d. Estimate P(X < 1.5), i.e., the proportion of all thickness values less than 1.5. [Hint: If you knew the values of m and s, you could calculate this probability. These values are not available, but they can be estimated.]
e. What is the estimated standard error of the estimator that you used in part (b)?
Answers
The coating thickness distribution is normal and has a mean value of 1.245.
What is mean?In statistics, in addition to the mode and median, the mean is one of the measures of central tendency. Simply put, the mean is the average of the values in the given set. It indicates that values in a particular data set are distributed equally. The three most frequently employed measures of central tendency are the mean, median, and mode.
To calculate the data's coating thickness, sample mean is employed.
The following formula is used to determine the mean value's point estimate:
\(\bar x = \frac{1}{n} \sum x_i\)
The typical probability plot shown there has a rather straight pattern. Assume right now that the coating thickness distribution is normal and has a mean value:
\(\bar x = \frac{19.92}{16}\\\\\bar x = 1.245\)
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You are curious whether acts of kindness have the same effect on college students. You convince your psychology professors to assign their students to do three random acts of kindness a week for a semester. At the end of the semester, you test the happiness of the students. You find that they have a mean happiness level of 6.55 (out of 8) with a standard deviation of 0.60. The national mean level of happiness for college students is 5.90 with an unknown standard deviation. Given this, would you start doing three random acts of kindness a week? - Yes
- No
- I'm not sure
(Note: Just answer honestly. This is an exploratory exercise. There are no right or wrong answers.) 2. Comparing the t and the z statistics You can use both the t statistic and the z statistic to test hypotheses about the mean of a population. The test that uses the t statistic is typically referred to as a t test, while the test that uses the z statistic is commonly called a z test. Which of the following statements are true of the t statistic? Check all that apply.
- The formula for the t statistic is t = (M-s)/SM. - The t statistic provides a relatively poor estimate of z with small sample sizes.
- If you know the population standard deviation, you should use the t statistic.
- The t statistic could be considered as an estimated z statistic. With the exception of whether the population standard deviation is known, the necessary assumptions for hypothesis tests with the t statistic and with the z statistic are ____
Answers
The following can be answered by the concept of Standard deviation.
Yes, I would start doing three random acts of kindness a week based on the results of the study. The mean happiness level of the students who did three random acts of kindness a week for a semester was 6.55, which is significantly higher than the national mean level of happiness for college students (5.90).
The standard deviation of the sample was also relatively small (0.60), suggesting that the effect of doing three random acts of kindness a week on happiness was consistent among the students. Therefore, it is reasonable to conclude that doing three random acts of kindness a week can lead to an increase in happiness among college students.
The necessary assumptions for hypothesis tests with the t statistic and with the z statistic are that the sample is a random sample from a normally distributed population, the population standard deviation is unknown for t-tests, and the sample size is sufficiently large for z-tests.
Additionally, the formula for the t statistic is t = (M-μ)/(s/√n), where M is the sample mean, μ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size. The t statistic provides a better estimate of the population parameter when the sample size is small, and the z statistic is more appropriate for larger sample sizes. If the population standard deviation is known, then the z statistic can be used instead of the t statistic.
Finally, the t statistic can be considered an estimated z statistic, but it incorporates additional uncertainty due to the small sample size.
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How can we ensure that we choose a sample of students that is representative of all 8:00 am classes that take place on a given morning?
Answers
By using a sampling technique we ensure that we choose a sample of students that is representative of all 8:00 am classes.
There are varieties of sampling strategies: chance sampling includes random selection, permitting you to make sturdy statistical inferences approximately the complete organization. Non-opportunity sampling entails non-random selection primarily based on convenience or different standards, permitting you to without problems gather records.
Random sampling is part of the sampling technique wherein every sample has an same possibility of being chosen. A sample chosen randomly is meant to be an unbiased representation of the overall population.
explanation;
we conclude the
6 buildings in the college 4 lecture halls in each building100 students in each lecture hallSince the students' lecture hallsare on different building the samples are
Dividing the students into groups, the students will be grouped by the buildings of their lecture halls.
The number of students in each building is:
There are 100 students in each building
Then select at random an equal proportion of student from each building let 20 students in each building.
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Please help me with this problem
Answers
Answer:
i cant see the pic please send a better one and i reloaded it and it still does not work
Step-by-step explanation:
If the length of the game room is 12 feet, what is the total square footage of the two rooms?
Answers
Answer:
Step-by-step explanation:
Well, all we have to do is square 12.
= \(12^{2}\)
= 12 x 12
= 144
The total square footage of the room is 144 feet.
I am not sure if this answer is correct, so please tell me if I am wrong. I hope this answer helped you! :)
The following chips are placed in a bucket: 4 red, 1 yellow, 5 blue, and 5 green. One chip is randomly selected from the bucket.
What is the probability that the chip is blue?
Answers
Answer:Red
Step-by-step explanation:
Answer:
Step-by-step explanation:
5 blue out of 15 chips
determine whether the geometric series is convergent or divergent. [infinity] 9(0.2)n − 1 n = 1
Answers
Given ;
9(0.2)n − 1 n = 1
The given geometric series is convergent.
convergent series:
Σ [from n=1 to infinity] 9(0.2)^(n-1)
To determine if a geometric series is convergent or divergent,
we need to look at the common ratio (r). In this case, r = 0.2.
A geometric series is convergent if the absolute value of the common ratio is less than 1 (|r| < 1) and divergent if the absolute value of the common ratio is greater than or equal to 1 (|r| >= 1).
Since |0.2| < 1, the given geometric series is convergent.
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2.333333 as a fraction
Answers
Answer:
2333333/1000000
Step-by-step explanation:
I hope this helps you :)
This is just probability, please help me !
Answers
Answer:
4/5
Step-by-step explanation:
Two are white.
10 total marbles.
10 - 2
8 are not white.
8/10
Simplify or reduce.
⇒ 4/5
Which of the following is a key property of the quadratic parent function?
OA. It is in quadrants III and IV.
OB. Its vertex is at the origin.
C. It is not a parabola.
OD. It is not a function.
Answers
The correct property of the quadratic parent function is given by:
B. Its vertex is at the origin.
What is the quadratic parent function?The quadratic parent function is modeled by:
y = x².
It has the vertex at the origin and increases to the left and to the right, into quadrants I and II, forming a function that is graphed by a parabola.
Hence the correct option regarding a property of the quadratic parent function is given by:
B. Its vertex is at the origin.
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PRECALCULUS PLEASE HELP
In the triangle, the length of side b is 7 the m∠A=60°. Find the exact lengths of a and c. Exact means you CANNOT give me decimals. You will need to give me either an integer a fraction or a radical. Use the unit circle. There will need to be two separate equations that you solve. Show all your work.
Answers
Answer:
Let a = x and c = y
Using the Law of Cosines:
x^2 = 7^2 + y^2 - 2(7)(y)cos60°
x^2 = 49 + y^2 - 14y
x^2 - y^2 + 14y = 49
(x - 7y)(x + 7y) = 49
x - 7y = ±7
x + 7y = ±7
Adding the two equations:
2x = ±14
x = ±7
Substituting x = ±7 into x - 7y = ±7:
±7 - 7y = ±7
-7y = 0
y = 0
Therefore, a = ±7 and c = 0
6 16 Next → Pretest: Scientific Notation Drag the tiles to the correct boxes to complete the pairs.. Particle Mass (grams) proton 1.6726 × 10-24 The table gives the masses of the three fundamental particles of an atom. Match each combination of particles with its total mass. Round E factors to four decimal places. 10-24 neutron 1.6749 × electron 9.108 × 10-28 two protons and one neutron one electron, one proton, and one neutron Mass 0-24 grams two electrons and one proton one proton and two neutrons Submit Test Particles F
Answers
We can drag the particles in mass/grams measurement to the corresponding descriptions as follows:
1. 1.6744 × 10⁻²⁴: Two electrons and 0ne proton
2. 5.021 × 10⁻²⁴: Two protons and one neutron
3. 5.0224 × 10⁻²⁴: One proton and two neutrons
4. 3.3484 × 10⁻²⁴: One electron, one proton, and one neutron
How to match the particlesTo match the measurements to the descriptions first note that one neutron is 1.6749 × 10⁻²⁴. One proton is equal to 1.6726 × 10⁻²⁴ and one electron is equal to 9.108 × 10⁻²⁸.
To obtain the right combinations, we have to add up the particles to arrive at the constituents. So, for the figure;
1.6744 × 10⁻²⁴, we would
Add 2 electrons and one proton
= 2(9.108 × 10⁻²⁸) + 1.6726 × 10⁻²⁴
= 1.6744 × 10⁻²⁴
The same applies to the other combinations.
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Manny has $12,000 and is saving for a used car that costs $15,000. He is able to save $300. 00 per month toward his car. Write and solve an inequality for this situation if m represents the number of months that Manny must save to buy the car.
Answers
The inequality representing this situation is m ≥ 10, indicating that Manny must save for 10 or more months to buy the car.
To represent Manny's saving situation for a used car, we can write and solve an inequality using the variable 'm' to represent the number of months he must save to buy the car.
Given information:
Manny has $12,000 initially.
The cost of the used car is $15,000.
Manny saves $300 per month.
The total amount Manny saves after 'm' months can be calculated as: $12,000 + ($300 * m).
To afford the car, the total amount Manny saves after 'm' months should be greater than or equal to the cost of the car: $12,000 + ($300 * m) ≥ $15,000.
Solve the inequality for 'm':
$12,000 + ($300 * m) ≥ $15,000.
Subtract $12,000 from both sides:
$300 * m ≥ $3,000.
Divide both sides by $300:
m ≥ 10.
Interpretation: Manny must save for at least 10 months to afford the used car. Therefore, the inequality representing this situation is m ≥ 10, indicating that Manny must save for 10 or more months to buy the car.
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.
The inverse f(x)= x^2 + 6x + 5 of the function is not a function. Which restriction of ensures that the inverse of is a function?
Answers
Alternatively, we could also restrict the domain of f(x) to a range that excludes the values of x that produce non-unique values of y, such as x = -3, which produces a value of y = 2 for f(x).
what is domain ?
In mathematics, the domain of a function is the set of all possible input values (also called independent variables) for which the function is defined and produces a valid output. It is the set of values that we are allowed to input into the function.
In the given question,
For the inverse of f(x) = x² + 6x + 5 to be a function, we need to ensure that it passes the vertical line test. In other words, for every value of x, the inverse function should produce only one unique value of y.
To ensure that the inverse of f(x) is a function, we need to restrict the domain of f(x) to a range that produces only one value of y for each value of x. This means that we need to make sure that f(x) is one-to-one, or injective, meaning that no two distinct values of x can produce the same value of y.
To check if f(x) is injective, we can use the discriminant of the quadratic equation x² + 6x + 5 = y, which is b² - 4ac, where a = 1, b = 6, and c = 5. The discriminant is:
b² - 4ac = 6² - 4(1)(5) = 16
Since the discriminant is positive, there are two distinct real roots of the quadratic equation, which means that f(x) is not injective and therefore does not have an inverse that is a function.
To ensure that the inverse of f(x) is a function, we need to restrict the domain of f(x) to a range that produces only one value of y for each value of x. One way to do this is to restrict the domain of f(x) to only include the values of x for which the discriminant is non-negative, meaning that the quadratic equation x² + 6x + 5 = y has real roots. This can be expressed as:
b² - 4ac >= 0
6² - 4(1)(5) >= 0
16 >= 0
This inequality is true for all values of x, which means that we can restrict the domain of f(x) to the entire real line to ensure that the inverse of f(x) is a function. Alternatively, we could also restrict the domain of f(x) to a range that excludes the values of x that produce non-unique values of y, such as x = -3, which produces a value of y = 2 for f(x).
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WAGES Mark has already earned money for mowing lawns over the summer when he takes a job at the local grocery store, earning $9.50 per hour. After working 16 hours at the grocery store, Mark has earned a total of $292. Write a linear equation to represent the amount of money m that Mark has earned this summer after working h hours at the grocery store.
PLEASE HELP!!!
Answers
Answer:
The linear equation to represent the amount of money m that Mark has earned this summer after working h hours at the grocery store is:
m = 9.5h + 140Step-by-step explanation:
GivenEarning per hour = $9.50,Worked hours = 16,Total earning = $292,Mark has earned some amount initially for mowing lawns.To findLinear equation of amount of money m after working h hoursSolutionMark has already earned some amount x. Lets find it using the number of hours worked and the total amount after this:
16*9.50 + x = 292152 + x = 292x = 292 - 152x = 140This amount represents an initial value or the y-intercept of the line and the payment per hour represents the slope of same line.
We know the equation of line in slope-intercept form:
y = mx + b, where y- line, m- slope, b - the y-interceptPlug in the values and variables to get the linear equation for the earned amount:
m = 9.5h + 140Mark's m earned this summer after working h hours at the grocery store is represented by the linear equation: m = 9.5h + 140.
What is meant by linear equation?A linear equation is an algebraic equation with only a constant and a first-order (linear) term of the form y = mx + b, where m is the slope and b is the y-intercept. The above is sometimes referred to as a "linear equation of two variables," where y and x are the variables. Linear equations are degree 1 equations. It is the straight line equation. The standard form of a linear equation is ax + by + c = 0, where a and b are both zeros.Given
Earning per hour = $9.50,
Worked hours = 16,
Total earning = $292,
Mark has earned some amount initially for mowing lawns.
Linear equation of amount of money m after working h hours
Mark has already earned some amount x.
Lets find it using the number of hours worked and the total amount after this:
16 × 9.50 + x = 292
Simplifying the above equation then we get,
152 + x = 292
x = 292 - 152
x = 140
This amount represents the line's initial value or y-intercept, and the payment per hour represents the slope of the same line.
The slope-intercept equation of a line is as follows:
y = mx + b, where y is the line, m is the slope, and b is the y-intercept
To obtain the linear equation for the earned amount, enter the values and variables as follows:
∴ m = 9.5h + 140
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Make m as the subject of
C/m+n=d/m-n
Answers
Answer: there is no answer
Step-by-step explanation: if the whole sentence is letters it is not answerable
the midterm exam scores obtained by boys and girls in a class are listed in the table below: what does the circled section represent?
Answers
12 boys scored 8 points in the exam.
It is given that the table shows the midterm exam results for boys and girls in a class.
As we can see clearly in the table in the first column we have the number of boys, in the second column we have their exam score and in the last column, we have number of the girls.
As we can see 12 is encircled in the first column and in the same row and second column there is 8 present which means 12 boys scored 8 points in the exam.
Therefore, 12 boys scored 8 points in the exam.
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Given question is incomplete, the complete question is given below:
the midterm exam scores obtained by boys and girls in a class are listed in the table below:
what does the circled section represent?
3 more than the quotient of 18 and x
Answers
Answer:
18/x + 3
Step-by-step explanation:
This can be written algebraically as 18/x +3. This question is a little unclear, but I think it’s asking to write it as an equation
Directions: Determine if the equations are parallel, perpendicular or neither.
11. x + y = 8 and y = -x – 1
X
-X
Answers
Suppose that 12 inches of wire costs 48 cents.
At the same rate, how much (in cents) will 6 inches of wire cost?
Answers
Answer:
Answer : 24
Step-by-step explanation:
Answer:
24
Step-by-step explanation:
obviously,you would have to do 48÷12=4
meaning 4×6=24
A set of blocks contains blocks of heights 1, 2, and 4 centimeters. Imagine constructing towers by piling blocks of different heights directly on top of one another. (A tower of height 6 cm could be obtained using six 1 cm blocks, three 2 cm blocks, one 2 cm block with one 4 cm block on top, one 4 cm block with one 2 cm block on top, and so forth.) Lett, be the number of ways to construct a tower of height n cm using blocks from the set. (Assume an unlimited supply of blocks of each size.) Use recursive thinking to obtain a recurrence relation for ty, ty, tzo Imagine a tower of height k cm. Either the bottom block has height 1 cm or it has height 2 cm or it has height cm. If the bottom block has height 1 cm, then the remaining blocks make up a tower of height x cm. By definition of t, there are tk-1 such towers. If the bottom block has height 2 cm, then the remaining blocks make up a tower of height x cm. By definition of there are x cm, then the remaining blocks make tx-2 such towers. If the bottom block has height such towers up a tower of height x cm. By definition of there are 1 Select X Therefore, for each integer, n 25,
Answers
Answer: Based on the problem statement, we can define a recurrence relation as follows:
t(n) = t(n-1) + t(n-2) + t(n-4)
This means that the number of ways to construct a tower of height n cm can be obtained by considering the possible heights of the bottom block in the tower. If the bottom block has height 1 cm, then the remaining blocks make up a tower of height (n-1) cm, for which there are t(n-1) ways to construct it. If the bottom block has height 2 cm, then the remaining blocks make up a tower of height (n-2) cm, for which there are t(n-2) ways to construct it. If the bottom block has height 4 cm, then the remaining blocks make up a tower of height (n-4) cm, for which there are t(n-4) ways to construct it.
Since we are assuming an unlimited supply of blocks of each size, we can use these blocks repeatedly to construct towers of different heights. Also, we can use dynamic programming to compute the values of t(n) for each integer n from 1 to 25, by using the recurrence relation above and the base cases:
t(0) = 1 (there is only one way to construct a tower of height 0 cm, which is to not use any blocks)
t(n) = 0 for n < 0 (there is no way to construct a tower of negative height)
Using these, we can compute the values of t(n) for n = 1, 2, ..., 25, as follows:
t(0) = 1
t(1) = t(0) = 1
t(2) = t(1) + t(0) = 2
t(3) = t(2) + t(1) = 3
t(4) = t(3) + t(2) + t(0) = 6
t(5) = t(4) + t(3) + t(1) = 10
t(6) = t(5) + t(4) + t(2) = 19
t(7) = t(6) + t(5) + t(3) = 32
t(8) = t(7) + t(6) + t(4) = 61
t(9) = t(8) + t(7) + t(5) = 104
t(10) = t(9) + t(8) + t(6) = 195
t(11) = t(10) + t(9) + t(7) = 332
t(12) = t(11) + t(10) + t(8) = 626
t(13) = t(12) + t(11) + t(9) = 1065
t(14) = t(13) + t(12) + t(10) = 2002
t(15) = t(14) + t(13) + t(11) = 3405
t(16) = t(15) + t(14) + t(12) = 6403
t(17) = t(16) + t(15) + t(13) = 10946
t(18) = t(17) + t(16) + t(14) = 20618
t(19) = t(18) + t(17) + t(15) = 350
Step-by-step explanation:
HElP MEh PleZ
A scuba diver descends below the surface of a lake at a rate of 12 feet per minute. What integer represents the depth of the diver after 4 minutes?
Answers
Since this scuba diver descends below the surface of a lake at a rate of 12 feet per minute, an integer which represents the depth of the diver after four (4) minutes is 48 feet.
What is the rate of change?The rate of change can be defined as a type of function that describes the average rate at which a quantity decreases or increases with respect to another quantity.
Mathematically, the rate of change for the depth covered by this scuba can be calculated by using this formula;
Rate of change = Depth/time
Making depth the subject of formula, we have:
Depth = Rate of change × time
Substituting the given parameters into the formula, we have;
Depth = 12 × 4
Depth = 48 feet.
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Suppose you have a data set with 9 distinct observations. if the largest number in the data set increases the standard deviation will:___________
Answers
If the largest number in the data set increases the standard deviation will increase.
Let us suppose that we are given a data set with 9 distinct observations.
It is given that the largest number in the data set increases.
We know that,
Standard deviation = \(\sqrt{ \frac{sum[(X-m)^{2}]}{N}}\)
Where,
X = Element of data set
m = mean of the data set
N = number of elements in the data set
Let us consider our data set to be:-
1,2,3,4,5,6,7,8,9
Here,
Mean, m = (1+2+3+4+5+6+7+8+9)/9 = 45/9 = 5
Hence,
\(sum(X-m)^{2}=(1-5)^{2}+(2-5)^{2}+(3-5)^{2}+(4-5)^{2}+(5-5)^{0}+(6-5)^{2}+(7-5)^{2}+(8-5)^{2}+(9-5)^{2} = 16+9+4+1+0+1+4+9+16=60\)
N = 9
Hence
Standard deviation = \(\sqrt{\frac{60}{9} }=\sqrt{\frac{20}{3} } =2.58 (approximately)\)
Now, let the largest element of the set that is 9 increases to 18.
Hence,
Mean , m = (1 +2+3+4+5+6+7+8+18)/9 = 54/9 = 6
N = 9
\(sum(X-m)^{2}=(1-6)^{2}+(2-6)^{2}+(3-6)^{2}+(4-6)^{2}+(5-6)^{0}+(6-6)^{2}+(7-6)^{2}+(8-6)^{2}+(18-6)^{2} = 25+16+9+4+1+0+1+4+144=206\)
Standard deviation = \(\sqrt{\frac{206}{9} }=4.78(approximately)\)
Hence, we can clearly see that the standard deviation increases.
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Can someone help me with this math homework please!
Answers
Given:
The sequence is:
-3, 5, -7, 9, -11,...
To find:
The correct statement about the given sequence.
Solution:
We have,
-3, 5, -7, 9, -11,...
Here, "..." means there are more terms in the sequence. So, the sequence has more than 5 terms and option A is incorrect.
From the given sequence it is clear that the 4th term is 9. So, option B is correct.
Clearly, 5th terms is -11. So, \(f(5)=-11\) and option C is incorrect.
The domain of a sequence is always the set of natural numbers. So, option D is correct.
In the given sequence the 4th term is 9. It means the point (4,9) lies on the graph of the sequence. So, option E is correct.
Therefore, the correct options are B, D, E.
Ray looked up prices of thirteen used Chevrolet HHR “retro” trucks in the classified ads and found these prices: $8,500,$8,500,$8,500,$9,900, $10,800, $10,800, $11,000, $12,500, $12,500,$13,000,$13,000, $14,500,and $23,000. (SHOW WORK)
a. Make a frequency table for this data set.
b. Find the mean. Round to the nearest dollar.
c. Find the median.
d. Find the mode.
e. Find the range.
f. Find the four quartiles.
g. Find the interquartile range.
h. Find the boundary for the upper outliers.
i. Find the boundary for the lower outliers.
j. How many outliers are there?
Answers
$8,500 - 2, $9,900 - 1, $10,800 - 2, $11,000 - 1, $12,500 - 2, $13,000 - 2, $14,500 - 1, and $23,000 - 1 are the frequencies shown in this data set's frequency table.
What does "mean" signify in math?In mathematics, the mean seems to be the average of a set of data, which is calculated by adding all the integers and then by dividing the result by the total number of numbers. For instance, the mean for the collection of values 8, 9, 5, 6, 7 is 7, since 8 + 9 + 5 + 6 + 7 = 35, and 35/5 = 7.
a. Frequency table:
$8,500 - 2
$9,900 - 1
$10,800 - 2
$11,000 - 1
$12,500 - 2
$13,000 - 2
$14,500 - 1
$23,000 - 1
b. To find the mean, we add up all the prices and divide by the number of trucks:
(8,500 + 8,500 + 8,500 + 9,900 + 10,800 + 10,800 + 11,000 + 12,500 + 12,500 + 13,000 + 13,000 + 14,500 + 23,000) / 13 = $12,269.23
c. To find the median, we first order the prices from least to greatest:
\(8,500, 8,500, 8,500, 9,900, 10,800, 10,800, 11,000, 12,500, 12,500, 13,000, 13,000, 14,500, 23,000\)
Since there are an odd number of prices, the median is the middle value, which is $12,500.
d. The mode is the value that appears the most frequently. Here, the mode is $8,500.
e. The range is the difference between the highest and lowest prices:
23,000 - 8,500 = $14,500
f. To find the quartiles, we first order the prices from least to greatest:
\(8,500, 8,500, 8,500, 9,900, 10,800, 10,800, 11,000, 12,500, 12,500, 13,000, 13,000, 14,500, 23,000\)
The median of the bottom half of the data constitutes the first quartile (Q1). In this case, it is the median of the first 8 values, which is $9,900.
The second quartile (Q2) is the median, which is $12,500.
The third quartile (Q3) is the median of the upper half of the data. In this case, it is the median of the last 8 values, which is $13,000.
g. The interquartile range is the difference between the third and first quartiles:
Q3 - Q1 = $13,000 - $9,900 = $3,100
h. To find the boundary for the upper outliers, we use the formula Q3 + (1.5 * IQR) = $13,000 + (1.5 * $3,100) = $19,650
i. To find the boundary for the lower outliers, we use the formula Q1 - (1.5 * IQR) = $9,900 - (1.5 * $3,100) = $4,650
j. An outlier is a value that is more than 1.5 times the interquartile range above Q3 or below Q1. In this case, the only outlier is $23,000
Please note that the quartiles, outliers and interquartile range are calculated using the interquartile range formula, this is one of the methods to identify them.
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A pyrmaid has a square base with sides 8 mm long and a height of 15.2 mm. Calculate the volume of the pyramid; round to the nearest tenth if necessary.
Answers
The volume of the pyramid with an 8 mm square base and a height of 15.2 mm is approximately 204.8 cubic millimeters.
To calculate the volume of the pyramid, we can use the formula V = (1/3) * base area * height. Since the base of the pyramid is a square, the area of the base is found by squaring the length of the side.
Given that the side length of the square base is 8 mm, the base area is (8 mm)^2 = 64 square mm. The height of the pyramid is given as 15.2 mm.
Now we can substitute the values into the volume formula: V = (1/3) * 64 square mm * 15.2 mm.
Calculating this, we get V = (1/3) * 972.8 cubic mm. Simplifying further, V = 324.27 cubic mm.
Rounding this to the nearest tenth, the volume of the pyramid is approximately 204.8 cubic millimeters.
Therefore, the volume of the pyramid is 204.8 cubic millimeters.
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answer two questions about systems aaa and bbb: system aaa \text{\quad}start text, end text system bbb \begin{cases}x-4y
Answers
Two questions about systems aaa and bbb, the given system is:
System aaa: x = -1/3 and y = -7/3
System bbb: x = -1/3 and y = -7/3
To answer two questions about systems aaa and bbb, let's first clarify the given system:
System aaa:
x - 4y = 9
System bbb:
2x + y = -3
Question 1: Solve system aaa.
To solve system aaa, we'll use the method of substitution:
Step 1: Solve one equation for one variable.
From the first equation in system aaa, we can isolate x:
x = 4y + 9
Step 2: Substitute the expression from Step 1 into the other equation.
Substitute the expression for x in the second equation of system aaa:
2(4y + 9) + y = -3
Step 3: Simplify and solve for y.
8y + 18 + y = -3
9y + 18 = -3
9y = -3 - 18
9y = -21
y = -21/9
y = -7/3
Step 4: Substitute the value of y into the expression for x.
Using the first equation in system aaa:
x - 4(-7/3) = 9
x + 28/3 = 9
x = 9 - 28/3
x = (27 - 28)/3
x = -1/3
Therefore, the solution to system aaa is x = -1/3 and y = -7/3.
Question 2: Solve system bbb.
To solve system bbb, we'll use the method of substitution:
Step 1: Solve one equation for one variable.
From the second equation in system bbb, we can isolate y:
y = -2x - 3
Step 2: Substitute the expression from Step 1 into the other equation.
Substitute the expression for y in the first equation of system bbb:
x - 4(-2x - 3) = 9
Step 3: Simplify and solve for x.
x + 8x + 12 = 9
9x + 12 = 9
9x = 9 - 12
9x = -3
x = -3/9
x = -1/3
Step 4: Substitute the value of x into the expression for y.
Using the second equation in system bbb:
y = -2(-1/3) - 3
y = 2/3 - 3
y = 2/3 - 9/3
y = -7/3
Therefore, the solution to system bbb is x = -1/3 and y = -7/3.
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(a) Show that the vectors u1 = (2, 0, 3), u2 = (−3, 0, 2) and u3 = (0, 7, 0) form an orthogonal basis for R 3 .(b) Write v = (1, 2, 3) as a linear combination of u1 = (2, 0, 3), u2 = (−3, 0, 2) and u3 = (0, 7, 0).
Answers
Main Answer:The linear combination of v = (13/14)u1 + (2/7)u2 + (47/14)u3
Supporting Question and Answer:
How can we express a vector as a linear combination of vectors using a system of equations?
To express a vector as a linear combination of vectors using a system of equations, we need to find the coefficients that multiply each given vector to obtain the desired vector. This can be done by setting up a system of equations, where each equation corresponds to the components of the vectors involved.
Body of the Solution:
(a) To show that the vectors u1 = (2, 0, 3), u2 = (-3, 0, 2), and u3 = (0, 7, 0) form an orthogonal basis for R^3, we need to demonstrate two conditions: orthogonality and linear independence.
Orthogonality: We need to show that each pair of vectors is orthogonal, meaning their dot product is zero.u1 · u2 = (2)(-3) + (0)(0) + (3)(2) = -6 + 0 + 6 = 0
u1 · u3 = (2)(0) + (0)(7) + (3)(0) = 0 + 0 + 0 = 0
u2 · u3 = (-3)(0) + (0)(7) + (2)(0) = 0 + 0 + 0 = 0
Since the dot product of every pair of vectors is zero, they are orthogonal.
2.Linear Independence: We need to show that the vectors u1, u2, and u3 are linearly independent, meaning that no vector can be written as a linear combination of the other vectors.
We can determine linear independence by forming a matrix with the vectors as its columns and performing row operations to check if the matrix can be reduced to the identity matrix.
[A | I] = [u1 | u2 | u3 | I] =
[2 -3 0 | 1 0 0]
[0 0 7 | 0 1 0]
[3 2 0 | 0 0 1]
Performing row operations:
R3 - (3/2)R1 -> R3
R1 <-> R2
[1 0 0 | -3/2 1 0]
[0 1 0 | 0 1 0]
[0 0 7 | 0 0 1]
Since we can obtain the identity matrix on the left side, the vectors u1, u2, and u3 are linearly independent.
Therefore, the vectors u1 = (2, 0, 3), u2 = (-3, 0, 2), and u3 = (0, 7, 0) form an orthogonal basis for R^3.
(b) To write v = (1, 2, 3) as a linear combination of u1, u2, and u3, we need to find the coefficients x, y, and z such that:
v = xu1 + yu2 + z*u3
Substituting the given vectors and coefficients:
(1, 2, 3) = x(2, 0, 3) + y(-3, 0, 2) + z(0, 7, 0)
Simplifying the equation component-wise:
1 = 2x - 3y
2 = 7y
3 = 3x + 2y
From the second equation, we can solve for y:
y = 2/7
Substituting y into the first equation:
1 = 2x - 3(2/7)
1 = 2x - 6/7
7 = 14x - 6
14x = 13
x = 13/14
Substituting the found values of x and y into the third equation
3 = 3(13/14) + 2(2/7)
3 = 39/14 + 4/7
3 = 39/14 + 8/14
3 = 47/14
Therefore, we have determined the values of x, y, and z as follows:
x = 13/14
y = 2/7
z = 47/14
Thus, we can write the vector v = (1, 2, 3) as a linear combination of u1 = (2, 0, 3), u2 = (-3, 0, 2), and u3 = (0, 7, 0) as:
v = (13/14)u1 + (2/7)u2 + (47/14)u3
Therefore, v can be expressed as a linear combination of the given vectors.
Final Answer:Therefore,the linear combination of v = (13/14)u1 + (2/7)u2 + (47/14)u3
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The linear combination of v = (13/14)u1 + (2/7)u2 + (47/14)u3
To express a vector as a linear combination of vectors using a system of equations, we need to find the coefficients that multiply each given vector to obtain the desired vector. This can be done by setting up a system of equations, where each equation corresponds to the components of the vectors involved.
Body of the Solution:
(a) To show that the vectors u1 = (2, 0, 3), u2 = (-3, 0, 2), and u3 = (0, 7, 0) form an orthogonal basis for R^3, we need to demonstrate two conditions: orthogonality and linear independence.
Orthogonality: We need to show that each pair of vectors is orthogonal, meaning their dot product is zero.
u1 · u2 = (2)(-3) + (0)(0) + (3)(2) = -6 + 0 + 6 = 0
u1 · u3 = (2)(0) + (0)(7) + (3)(0) = 0 + 0 + 0 = 0
u2 · u3 = (-3)(0) + (0)(7) + (2)(0) = 0 + 0 + 0 = 0
Since the dot product of every pair of vectors is zero, they are orthogonal.
2.Linear Independence: We need to show that the vectors u1, u2, and u3 are linearly independent, meaning that no vector can be written as a linear combination of the other vectors.
We can determine linear independence by forming a matrix with the vectors as its columns and performing row operations to check if the matrix can be reduced to the identity matrix.
[A | I] = [u1 | u2 | u3 | I] =
[2 -3 0 | 1 0 0]
[0 0 7 | 0 1 0]
[3 2 0 | 0 0 1]
Performing row operations:
R3 - (3/2)R1 -> R3
R1 <-> R2
[1 0 0 | -3/2 1 0]
[0 1 0 | 0 1 0]
[0 0 7 | 0 0 1]
Since we can obtain the identity matrix on the left side, the vectors u1, u2, and u3 are linearly independent.
Therefore, the vectors u1 = (2, 0, 3), u2 = (-3, 0, 2), and u3 = (0, 7, 0) form an orthogonal basis for R^3.
(b) To write v = (1, 2, 3) as a linear combination of u1, u2, and u3, we need to find the coefficients x, y, and z such that:
v = xu1 + yu2 + z*u3
Substituting the given vectors and coefficients:
(1, 2, 3) = x(2, 0, 3) + y(-3, 0, 2) + z(0, 7, 0)
Simplifying the equation component-wise:
1 = 2x - 3y
2 = 7y
3 = 3x + 2y
From the second equation, we can solve for y:
y = 2/7
Substituting y into the first equation:
1 = 2x - 3(2/7)
1 = 2x - 6/7
7 = 14x - 6
14x = 13
x = 13/14
Substituting the found values of x and y into the third equation
3 = 3(13/14) + 2(2/7)
3 = 39/14 + 4/7
3 = 39/14 + 8/14
3 = 47/14
Therefore, we have determined the values of x, y, and z as follows:
x = 13/14
y = 2/7
z = 47/14
Thus, we can write the vector v = (1, 2, 3) as a linear combination of u1 = (2, 0, 3), u2 = (-3, 0, 2), and u3 = (0, 7, 0) as:
v = (13/14)u1 + (2/7)u2 + (47/14)u3
Therefore, v can be expressed as a linear combination of the given vectors.
Therefore, the linear combination of v = (13/14)u1 + (2/7)u2 + (47/14)u3
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