Answer:
160 nickels, 80 pennies
Step-by-step explanation:
160 nickels = $8
80 pennies = $0.80
$8 + $0.80 = $8.80
Troy puts $200.00 into an account to use for school expenses. The account earns 7% interest, compounded annually. How much will be in the account after 5 years? Use the formula A=P1+ r n nt, where A is the balance (final amount), P is the principal (starting amount), r is the interest rate expressed as a decimal, n is the number of times per year that the interest is compounded, and t is the time in years. Round your answer to the nearest cent.
Answer:
the final amount is = $280.51
Step-by-step explanation:
The standard formula for compound interest is given as;
A = P(1+r/n)^(nt) .....1
Given that;
Principal P = $200
Interest rate r = 7% = 0.07
Time t = 5 years
Final amount = A
Number of time compounded per year n = 1
Substituting the values;
A = 200(1+0.07/1)^(1×5)
A = 280.51
Therefore, the final amount is = $280.51
can someone help please and explain i’m gonna mark brainliest!!
Answer:
I believe it is C
Step-by-step explanation:
According to the 30-60-90 rules for the length of the sides. the short leg is x the long leg is xsquareroot3 and the hypotenuse is 2x. Following that rule, A is 3square root 3 and b is 3
what's the answer !!
\( \sqrt{15 \times 15} \)
Answer:
\( \sqrt{15 \times 15} \)
:\( \sqrt{15²} \)=±15 is a right answer
umm... lemme tryy
\(\sqrt{15 \times 15} = \sqrt{152} = \sqrt{15} \)
I hope my answer is correct HUHUHU fingers crossed
HELP ASAPPPP
In two sample surveys, 125 people were asked about their favorite fruit. In the first survey, 40 people chose apples, 64 chose oranges, and 21 chose bananas. In the second, 43 chose apples, 63 chose oranges, and 19 chose bananas. Marianne inferred that most people prefer oranges. Is this inference true based on the data? Explain.
Answer:
More than half of the people surveyed in each sample chose oranges as their favorite fruit. Since most people in each sample chose oranges, it is likely that oranges are the favorite fruit of the entire population.
I hope this helps you
Step-by-step explanation:
The surface area of a sphere is 200. 96 square centimeters. What is the approximate volume of the sphere? Use 3. 14 for pi
Answer:
200.96 (pi)= 631.33
200.96. (3.14)=631.01
Step-by-step explanation:
when you use (pi) the answer is 631.33
but if you use (3.14) the answer is 631.01
that's mean than the answer change
The population of a county is growing at a rate of 9% per year, compounded continuously. If the county currently has 14,934 people, how many will there be in this county in 10 years? Round to the nearest person.
3) In June a gallon of gasoline cost $2.75. In July the price increased to
$2.92. What is the percent of increase of a gallon of gasoline from June to
July? Round to the nearest tenth.
Answer:
6.2%
Step-by-step explanation:
Hard to explain, but easy. Take your new value, and subtract your initial (starting) value from the new value. In this case, it would be 2.95-2.75= 17. You now can take you difference (17) and divide it by the initial value, so 17/2.75. This gives you 6.181818182. Round to 6.2%
26 - (2c + 4) = 2(C+6) +C
what is c
Answer:
c = 2
Step-by-step explanation:
26 - (2c + 4) = 2(c+6) + c
26 - 2c + 4 = 2c + 12 + c
26 - 4 - 12 = 2c + 2c + c
10 = 5c
2 = c
find the nth term of the sequence 7, 17, 27, 37
Answer:
47
Step-by-step explanation:
7 plus 10 = 17, 17 plus 10, 27 and it goes on. pls give brainliest
Answer:
47
Step-by-step explanation:
10+10 ang bawat idadag sa bawat space
The box plots display measures from data collected when 20 people were asked about their wait time at a drive-thru restaurant window.
A horizontal line starting at 0, with tick marks every one-half unit up to 32. The line is labeled Wait Time In Minutes. The box extends from 8.5 to 15.5 on the number line. A line in the box is at 12. The lines outside the box end at 3 and 27. The graph is titled Super Fast Food.
A horizontal line starting at 0, with tick marks every one-half unit up to 32. The line is labeled Wait Time In Minutes. The box extends from 9.5 to 24 on the number line. A line in the box is at 15.5. The lines outside the box end at 2 and 30. The graph is titled Burger Quick.
Which drive-thru typically has more wait time, and why?
Burger Quick, because it has a larger median
Burger Quick, because it has a larger mean
Super Fast Food, because it has a larger median
Super Fast Food, because it has a larger mean
Question 2
The box plot represents the number of tickets sold for a school dance.
A horizontal line labeled Number of Tickets sold that starts at 8, with tick marks every one unit up to 30. The graph is titled Tickets Sold for A Dance. The box extends from 17 to 21 on the number line. A line in the box is at 19. The lines outside the box end at 10 and 27.
Which of the following is the appropriate measure of center for the data, and what is its value?
The mean is the best measure of center, and it equals 19.
The median is the best measure of center, and it equals 4.
The median is the best measure of center, and it equals 19.
The mean is the best measure of center, and it equals 4.
Question 3
A charity needs to report its typical donations received. The following is a list of the donations from one week. A histogram is provided to display the data.
5, 5, 6, 8, 10, 15, 18, 20, 20, 20, 20, 20, 20
A graph titled Donations to Charity in Dollars. The x-axis is labeled 1 to 5, 6 to 10, 11 to 15, and 16 to 20. The y-axis is labeled Frequency. There is a shaded bar up to 2 above 1 to 5, up to 3 above 6 to 10, up to 1 above 11 to 15, and up to 7 above 16 to 20.
Which measure of variability should the charity use to accurately represent the data? Explain your answer.
The range of 13 is the most accurate to use, since the data is skewed.
The IQR of 13 is the most accurate to use, since the data is skewed.
The range of 20 is the most accurate to use to show that they have plenty of money.
The IQR of 20 is the most accurate to use to show that they need more money.
Question 4
The circle graph describes the distribution of preferred transportation methods from a sample of 400 randomly selected San Francisco residents.
circle graph titled San Francisco Residents' Transportation with five sections labeled walk 40 percent, bicycle 8 percent, streetcar 15 percent, bus 10 percent, and cable car 27 percent
Which of the following conclusions can we draw from the circle graph?
Together, Streetcar and Cable Car are the preferred transportation for 168 residents.
Together, Walk and Streetcar are the preferred transportation for 55 residents.
Bus is the preferred transportation for 45 residents.
Bicycle is the preferred transportation for 50 residents.
Question 5
The line plot displays the number of roses purchased per day at a grocery store.
A horizontal line starting at 1 with tick marks every one unit up to 10. The line is labeled Number of Rose Bouquets, and the graph is titled Roses Purchased Per Day. There is one dot above 1 and 2. There are two dots above 8. There are three dots above 6, 7, and 9.
Which of the following is the best measure of variability for the data, and what is its value?
The range is the best measure of variability, and it equals 8.
The range is the best measure of variability, and it equals 2.5.
The IQR is the best measure of variability, and it equals 8.
The IQR is the best measure of variability, and it equals 2.5.
Question 7
At a recent baseball game of 5,000 in attendance, 150 people were asked what they prefer on a hot dog. The results are shown.
Ketchup Mustard Chili
63 27 60
Based on the data in this sample, how many of the people in attendance would prefer mustard on a hot dog?
900
2,000
2,100
4,000
Question 8
The number of milligrams of Vitamin C from 100 different gummy vitamins sold in the world was collected.
Which graphical representation would be most appropriate for the data, and why?
Box plot, because the median can easily be determined from the large set of data
Stem-and-leaf plot, because you can see the shape of the data
Histogram, because it shows each individual data point
Bar chart, because the data is categorical
Burger Quick, because it has a larger median.
How to explain the informationBurger Quick typically has more wait time because it has a larger range (30-2=28) compared to Super Fast Food (27-3=24) and also has a larger interquartile range (IQR) (24-9.5=14.5) compared to Super Fast Food (15.5-8.5=7), indicating more variability in the wait times. The fact that Burger Quick's median (15.5) is larger than Super Fast Food's median (12) reinforces this conclusion.
Therefore, the correct answer is: Burger Quick, because it has a larger median.
Answer 2: The appropriate measure of center for this data is the median, which is represented by the line in the box at 19. This is because the data is skewed, with a longer tail to the right, and the median is less sensitive to extreme values compared to the mean. Therefore, the correct answer is: The median is the best measure of center, and it equals 19.
Answer 3: The measure of variability that the charity should use to accurately represent the data is the IQR (Interquartile Range), which is the range of the middle 50% of the data. The reason for this is that the data is skewed to the right, with many values concentrated in the upper range of the data, and the IQR is less sensitive to outliers compared to the range.
The IQR for this data is 20-10=10, which represents the spread of the middle 50% of the data. Therefore, the correct answer is: The IQR of 20 is the most accurate to use, since the data is skewed.
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is the product of 1 prime number and 1 composite number is a composite number
Order the numbers from least to greatest!
-1.6, -75%, 5/2, -7/8, 0.9, 6/5
Please I need help!
Answer: -0.875 -0.75 -1.6 0.9 1.2 2.5
Step-by-step explanation:
Math question answer please
+5(9/3)²
Answer:
45
Step-by-step explanation:
What is the product of (2x^2 + 4x-2) and (3x+6) ?
O A. 6x^3 +12x^2 +24x - 12
B. 6x^3 + 24x^2 - 18x-12
C. 6x^3 + 12x^2 - 6x - 12
D. 6x^3 + 24x^2 +18x -12
Answer: the answer is 6x³ + 24x² + 18x - 12!
Step-by-step explanation:
Find the value of the line integral. F · dr C (Hint: If F is conservative, the integration may be easier on an alternative path.) F(x,y) = yexyi + xexyj (a) r1(t) = ti − (t − 4)j, 0 ≤ t ≤ 4 (b) the closed path consisting of line segments from (0, 4) to (0, 0), from (0, 0) to (4, 0), and then from (4, 0) to (0, 4)
To find the value of the line integral, we need to integrate the dot product of the vector field F with the differential vector dr along path C.
(a) Using the parametric equation r1(t) = ti - (t-4)j, we can calculate dr/dt = i - j and substitute it into the line integral formula:
∫ F · dr = ∫ (yexyi + xexyj) · (i-j) dt
= ∫ (ye^(t-i) - xe^(t-i)) dt from t=0 to t=4
= [ye^(t-i) + xe^(t-i)] from t=0 to t=4
= (4e^3 - 4e^-1) + (0 - 0)
= 4e^3 - 4e^-1
(b) To use an alternative path for easier integration, we can check if the vector field F is conservative.
∂M/∂y = exy + xexy = ∂N/∂x
where F = M(x,y)i + N(x,y)j
Thus, F is conservative and we can use the path independence property of conservative vector fields.
Going from (0,4) to (0,0) to (4,0) to (0,4) is equivalent to going from (0,4) to (4,0) to (0,0) to (0,4) and back to the starting point.
Using Green's theorem, we have:
∫ F · dr = ∫ M dy - ∫ N dx = ∫∫ (∂N/∂x - ∂M/∂y) dA
= ∫∫ (exy + xexy - exy - xexy) dA
= 0
Therefore, the value of the line integral along the closed path is zero.
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how far is the point (8,6) from the origin
(8 points) Find the volume of the solid in R3 bounded by y = x², x = y2, z = x + y + 9, and z = 0. X= = V=
The volume of the solid bounded by the given surfaces is 49/30 cubic units.
To find the volume of the solid bounded by the given surfaces, we need to determine the limits of integration for each variable. Let's analyze the given surfaces one by one.
The curve y = x²:
Since x = y² is another bounding surface, we can find the limits of integration by solving the system of equations y = x² and x = y².
Substituting x = y² into y = x², we get:
y = (y²)²
y = y⁴
y⁴ - y = 0
y(y³ - 1) = 0
This equation has two solutions: y = 0 and y = 1.
The curve x = y²:
Substituting x = y² into z = x + y + 4, we have:
z = y² + y + 4
Now we need to find the limits of integration for y. For that, we consider the region between the curves y = 0 and y = 1.
The limits of integration for y are 0 and 1.
The surface z = 0:
This surface represents the xy-plane and acts as the lower bound for the volume.
Therefore, the limits of integration for z are 0 and z = y² + y + 4.
To calculate the volume, we integrate the constant 1 with respect to x, y, and z over the given bounds:
V = ∫∫∫ dV
V = ∫[0,1]∫[0,y²]∫[0,y²+y+4] dz dx dy
V = ∫[0,1] (y² + y + 4 - 0) [y²] dy
V = ∫[0,1] (y⁴ + y³ + 4y²) dy
V = (1/5)y⁵ + (1/4)y⁴ + (4/3)y³ |[0,1]
V = (1/5)(1)⁵ + (1/4)(1)⁴ + (4/3)(1)³ - (1/5)(0)⁵ - (1/4)(0)⁴ - (4/3)(0)³
V = 1/5 + 1/4 + 4/3
V = 3/60 + 15/60 + 80/60
V = 98/60
Simplifying the fraction, we get:
V = 49/30
Therefore, the volume of the solid bounded by the given surfaces is 49/30 cubic units.
Incomplete question:
Find the volume of the solid in R3 bounded by y = x², x = y², z = x + y + 4, and z = 0.
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During summer weekdays, boats arrive at the inlet drawbridge according to the Poisson distribution at a rate of 4 per hour. Answer the next questions, Problem 6 parts a - d, below. Enter your answers in the space provided. Express your answer as a number to 4 decimal places using standard rounding rules. Attach your Excel file in Problem 6e. Problem 6a. What is the probability that no boats arrive in a 2-hour period? Problem 6b. What is the probability that 1 boat arrives in a 2-hour period? Problem 6a. What is the probability that no boats arrive in a 2-hour period? Problem 6b. What is the probability that 1 boat arrives in a 2-hour period? Problem 6c. What is the probability that 2 boats arrive in a 2-hour period? Problem 6d. What is the probability that 2 or more boats arrive in a 2- hour period?
a. The probability that no boats arrive in a 2-hour period is approximately 0.0003.
b. The probability that 1 boat arrives in a 2-hour period is approximately 0.0023.
c. The probability that 2 boats arrive in a 2-hour period is approximately 0.0466.
d. The probability that 2 or more boats arrive in a 2-hour period is approximately 0.9511.
What is probability?Probability is a way to gauge how likely something is to happen. Many things are difficult to forecast with absolute confidence. Using it, we can only make predictions about the likelihood of an event happening, or how likely it is.
Given that boats arrive at the inlet drawbridge according to a Poisson distribution with a rate of 4 per hour, we can use the Poisson probability formula to calculate the probabilities.
The Poisson probability mass function is given by:
P(x; λ) = \((e^{(-\lambda)} * \lambda^x) / x!\)
where x is the number of events, λ is the average rate of events.
(a) To find the probability that no boats arrive in a 2-hour period, we can calculate P(0; λ), where λ is the average rate of events in a 2-hour period. Since the rate is 4 boats per hour, the average rate in a 2-hour period is λ = 4 * 2 = 8.
P(0; 8) = \((e^{(-8)} * 8^0) / 0! = 8e^{(-8)}\) ≈ 0.0003
The probability that no boats arrive in a 2-hour period is approximately 0.0003.
(b) To find the probability that 1 boat arrives in a 2-hour period, we can calculate P(1; λ), where λ is the average rate of events in a 2-hour period (λ = 8).
P(1; 8) = \((e^{(-8)} * 8^1) / 1! = 8e^{(-8)}\) ≈ 0.0023
The probability that 1 boat arrives in a 2-hour period is approximately 0.0023.
(c) To find the probability that 2 boats arrive in a 2-hour period, we can calculate P(2; λ), where λ is the average rate of events in a 2-hour period (λ = 8).
P(2; 8) = \((e^{(-8)} * 8^2) / 2! = (64/2) * e^{(-8)}\) ≈ 0.0466
The probability that 2 boats arrive in a 2-hour period is approximately 0.0466.
(d) To find the probability that 2 or more boats arrive in a 2-hour period, we need to calculate the complement of the probability that 0 or 1 boat arrives.
P(2 or more; 8) = 1 - (P(0; 8) + P(1; 8))
P(2 or more; 8) \(= 1 - (e^(-8) + 8e^{(-8)})\) ≈ 0.9511
The probability that 2 or more boats arrive in a 2-hour period is approximately 0.9511.
Please note that the above probabilities are calculated based on the assumption of a Poisson distribution with a rate of 4 boats per hour.
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solve for x
4x + 9
16x - 9
Answer:
9
Step-by-step explanation:
Vertical lines are parallel so the given interior angles are supplementary and their sum is equal to 180:
4x + 9 + 16x - 9 = 180 add like terms
20x = 180 divide both sides by 20
x = 9
Elyse has a gift card to a local movie theater. the graph shows the amount of money remaining on her gift card based on the number of movies she has seen.
a. write an equation to represent the situation.
b. interpret the slope and y-intercept in the context of the situation.
a. The equation to represent the situation is y = -12x + 120, where x is the number of movies and y is the amount of money remaining on the gift card.
What is money?Money is a medium of exchange that is widely accepted as a way to pay for goods and services or to settle debts. Money also serves as a store of value, providing a way for people to save for the future. Money is generally created through government-backed fiat currencies, such as the U.S. dollar, which are issued and regulated by central banks. Money can also be created in the form of crypto-currencies, such as Bitcoin, which are not issued by any single government or central bank. Money is essential for economic growth and stability, as it allows for efficient exchanges of goods and services. Money can also be a source of financial security, providing people with a way to manage their finances and plan for the future.
b. The slope of -12 indicates that for every movie that Elyse sees, she will spend $12 from her gift card. The y-intercept of 120 indicates that if Elyse has not seen any movies, she will have $120 remaining on her gift card.
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The situation is,
a) The line equation is \(y = -3x - 6\)
b) The line's y-intercept is -6, which indicates that when the amount of movies x = 0 , the amount on gift y = -6
c) The slope of the line is -3, indicating that as the number of movies x increases, the rate of change of the amount on the gift is declining.
What is an Equation of a line?a). The equation provides the line's slope.
Slope,
\(m=\frac{(y_2-y_1)}{(x_2-x_1)}\)
Changing the numbers indicated in the slope equation,
Slope,
\(m=\frac{(6-12)}{(4-2)}\)
Slope m = -6/2
Slope m = -3
The slope is -3
The equation of the line is,
\(y - y_1 = m ( x - x_1 )\)
Substitute the given values in the equation,
\(y - 12 = -3 ( x - 2 )\)
Simplify the equation,
\(y - 12 = -3x + 6\)
Adding 12 on both sides
\(y = -3x - 6\)
The equation of line is \(y = -3x - 6\)
b). The y-intercept of the equation of line \(y = -3x - 6\) is \(-6\), when \(x=0\)
c). The slope of the line \(y = -3x - 6\) is \(m = -3\) and the value is decreasing
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The completr question and graph attached below,
c. What is the slope, and what does it mean in the context of the situation?
∠VTW≅∠UTW and ∠U≅∠V. Complete the proof that
VW
≅
UW
.
T
U
V
W
Answer: did you make that up ??
Step-by-step explanation:
Answer:
A. Alternate interior
B. Transitive property
C. Converse alternate interior angles theorem.
The idea that group work theories are appropriate for all clients all the time is a(n)? empirical truth professional tenet myth lie
The idea that group work theories are appropriate for all clients all the time is a myth (Third option).
About Group Process:
The term "group process" describes how people collaborate within an organization to complete tasks. Typically, organizations invest a lot of time and effort into defining and achieving goals, but little thought is usually given to how those goals are affecting the most valuable resource of a group - its people.
Since different clients have different demands and priorities, group work might not be suitable for many, but not for all. Different types of clients require different types of collaboration, type of group or individuals, and working strategies that work for their goals the best.
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13k + 6 + 2y + 9 + k
Please answer
Answer:
14 + 2 + 15
Step-by-step explanation:
Write a function that accepts a two-dimensional list as an argument and returns whether the list represents a magic square (either true or false).
The function would check if the given two-dimensional list represents a magic square and return True or False accordingly.
Below is a Python function that accepts a two-dimensional list as an argument and determines whether the list represents a magic square:
```python
def is_magic_square(square):
# Get the size of the square
n = len(square)
# Calculate the expected sum of each row, column, and diagonal
magic_sum = sum(square[0])
# Check rows
for row in square:
if sum(row) != magic_sum:
return False
# Check columns
for j in range(n):
col_sum = sum(square[i][j] for i in range(n))
if col_sum != magic_sum:
return False
# Check diagonals
diag_sum1 = sum(square[i][i] for i in range(n))
diag_sum2 = sum(square[i][n - i - 1] for i in range(n))
if diag_sum1 != magic_sum or diag_sum2 != magic_sum:
return False
return True
```
The `is_magic_square` function takes a two-dimensional list `square` as an argument. It first calculates the expected sum of each row, column, and diagonal by summing the elements in the first row (`square[0]`). Then it proceeds to check if the sum of each row, column, and both diagonals equals the calculated `magic_sum`. If any of these sums do not match `magic_sum`, the function returns `False`. If all sums match `magic_sum`, the function returns `True`, indicating that the input list represents a magic square.
You can call this function by passing your two-dimensional list as an argument, for example:
```python
my_square = [[2, 7, 6], [9, 5, 1], [4, 3, 8]]
result = is_magic_square(my_square)
print(result) # Output: True
```
Please note that the function assumes the input list is a square matrix, meaning it has the same number of rows and columns.
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If the range of the set of data below is 28, what is the missing number?
21, 26, 33, 35, 44, 47,
The missing number of the set is 19 or 49.
We are given that;
The number series 21, 26, 33, 35, 44, 47
Now,
The range is the difference between the maximum and minimum values in the data set. Here are the steps to find the missing number:
First, we need to identify the maximum and minimum values in the data set. The maximum value is 47 and the minimum value is 21.
Next, we need to subtract the minimum value from the maximum value to find the range. This gives us 47 - 21 = 26.
Since we are given that the range is 28, we need to find a number that would make the range 28. This means that we need to either increase the maximum value or decrease the minimum value by 2.
One possible way to do this is to replace 21 with 19. This would make the minimum value 19 and the range 47 - 19 = 28.
Another possible way to do this is to replace 47 with 49. This would make the maximum value 49 and the range 49 - 21 = 28.
Therefore, by the given range the answer will be 19 or 49.
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Select the correct solution for 12× = 60.
Given the expression:
\(\text{ 12x = 60}\)Let's determine the solution or the value of x.
Step 1: Applying the rule of the Inverse Property of Multiplication, we divide 60 by 12 to get the
value of x.
\(\text{ 12x = 60 }\rightarrow\text{ x = }\frac{60}{12}\)\(\text{ x = 5}\)Therefore, the solution to this expression is x = 5.
Find the LCM of 6, 24 and 32
Answer:
96
Step-by-step explanation:
25/35 Marks
Find the midpoint of A and B where A has coordinates (8, 2)
and B has coordinates (3,8).
Answer:
\(\displaystyle (\frac{11}{2},5)\)
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightAlgebra I
Coordinates (x, y)Midpoint Formula: \(\displaystyle (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})\)Step-by-step explanation:
Step 1: Define
Point A(8, 2) → x₁ = 8, y₁ = 2
Point B(3, 8) → x₂ = 3, y₂ = 8
Step 2: Find Midpoint
Simply plug in your coordinates into the midpoint formula to find midpoint
Substitute in points [Midpoint Formula]: \(\displaystyle (\frac{8+3}{2},\frac{2+8}{2})\)[Midpoint] [Fraction] Add: \(\displaystyle (\frac{11}{2},\frac{10}{2})\)[Midpoint] Divide: \(\displaystyle (\frac{11}{2},5)\)Consider this graph of a line.
The linear equation of the line shown in the graph will be 3y +x= 0.
What is the Linear equation?A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included. Sometimes, the aforementioned is referred to as a "linear equation of two variables," with y and x serving as the variables.
As we can observe in the given graph of lines,
the given line passes through the point (0, 0) origin and (-6, 2).
The slope-intercept form of the equation of a line
y=mx+b,
where m is the slope
b is the y-intercept
Since y-intercept is the value of y in line at x = 0
in our case, b = 0
since, slope = (y - y')/(x -x')
Slope = (2-0)/(-6-0)
Slope = 2/-6
Slope = -1/3
thus,
the equation of the line shown in the graph will be y = -1/3x
the equation of the line shown in the graph will be 3y +x= 0.
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The base of a solid S is the region enclosed by the graph of y=√ln(x), x=e, y=0. If the cross section of S perpendicular to the x-axis are squares, determine the volume V, of S.1) 1 cu. units.2) 13(e3−1) cu. units.3) 12 cu.units.4) 23 cu.units.5) 2(e3−1) cu.units.
The volume V of solid S is e - 1 cubic unit.
What is Volume?
Volume refers to the measure of three-dimensional space occupied by an object or a region. It quantifies the amount of space enclosed by the boundaries of an object or contained within a given region. In mathematical terms, volume is often calculated by integrating the cross-sectional areas of the object or region along a particular axis. Volume is typically expressed in cubic units, such as cubic meters (m^3) or cubic centimeters (cm^3). It is an essential concept in geometry, physics, engineering, and other scientific fields where the measurement of three-dimensional space is involved.
To find the volume of solid S, we need to integrate the areas of the cross sections perpendicular to the x-axis along the interval \([e, \infty).\)
The area of each square cross-section is equal to the square of the side length, which in this case is \(y = \sqrt{\ln(x)}.\)
Therefore, the volume V of solid S can be calculated as:
\(V = \int_{e}^{\infty} (\sqrt{\ln(x)})^2 dx\)
To evaluate this integral, we can simplify the expression:
\(V = \int_{e}^{\infty} \ln(x) dx\)
Using integration by parts, we let \(u = \ln(x)\)and dv = dx:
\(du = \frac{1}{x} dx\\v = x\)
Applying the integration by parts formula:
\(V = [uv] - \int v du= [x \ln(x)] - \int x \left(\frac{1}{x}\right) dx= x \ln(x) - \int dx= x \ln(x) - x + C\)
Evaluating the definite integral:
\(V = [x \ln(x) - x]_{e}^{\infty}= (\infty \cdot \ln(\infty) - \infty) - (e \cdot \ln(e) - e)= \infty - 0 - (1 - e)= e - 1\)
Therefore, the volume V of solid S is e - 1 cubic unit.
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