14. Find a counterexample for each of the following: a. If n is a natural number, then(n+5)/5 -n-1. b. If n is a natural number, then (n + 4)² = n² + 4².
Answers
n = 1 is a counterexample for both the given equations.
Here, we are given two equations, let us look at them one by one-
a. (n + 5) / 5 = n + 1
let us take n = 1
the, (1 + 5) / 5 = 1 + 1
6/ 5 = 2
This equation does not hold as 6/5 ≠ 2. Thus, n = 1 doesn't satisfy the given equation and hence is a counterexample.
b. (n + 4)² = n² + 4²
Let us take n = 1
(1 + 4)² = 1² + 4²
5² = 1 + 16
25 = 17
Again, this equation does not hold as 25 ≠ 17. Thus, n = 1 doesn't satisfy the given equation and hence is a counterexample.
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Related Questions
The weight of an object on moon is 1/6 of its weight on Earth. If an object weighs 1535 kg on Earth. How much would it weigh on the moon?
Answers
Answer:
255.8
Step-by-step explanation:
first
1/6*1535
=255.8
f(x) 10 3 5 2 Let glx) be a twice differentiable function defined by another differentiable function f(x), such that 9 (2) = 21+ J' f (t) dt: Selected values of f(x) are given in the table above
Answers
The derivative of the function g(x), which is defined by another differentiable function f(x), is g'(x) = 20 + 6x + 10x^2 + 4x^3.
f(x) 10 3 5 2 Let glx) be a twice differentiable function defined by another differentiable function f(x), such that 9 (2) = 21+ J' f (t) dt: Selected values of f(x) are given in the table above.
We can calculate g'(x) as follows:
g'(x) = 2f'(x) = 2(10 + 3x + 5x^2 + 2x^3) = 20 + 6x + 10x^2 + 4x^3
1. Start by taking the derivative of g(x) with respect to x. This is done using the chain rule, since g(x) is defined by another differentiable function f(x):
g'(x) = d/dx[21 + J' f(t) dt] = 2f'(x)
2. Now, substitute the function f(x) into the equation for g'(x):
g'(x) = 2(10 + 3x + 5x^2 + 2x^3) = 20 + 6x + 10x^2 + 4x^3
The derivative of the function g(x), which is defined by another differentiable function f(x), is g'(x) = 20 + 6x + 10x^2 + 4x^3.
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log8(_)-log8 7 = log8 5/7
fill in the blank (_)
Answers
\(\begin{array}{llll} \textit{Logarithm of rationals} \\\\ \log_a\left( \frac{x}{y}\right)\implies \log_a(x)-\log_a(y) \end{array} \\\\[-0.35em] ~\dotfill\\\\ \log_8(x)-\log_8(7)=\log_8\left( \cfrac{5}{7} \right)\implies \log_8(\stackrel{x }{5})-\log_8(7)=\log_8\left( \cfrac{5}{7} \right)\)
The port of South Louisiana, located along miles of the Mississippi River between New Orleans and Baton Rouge, is the largest bulk cargo port in the world. The U.S. Army Corps of Engineers reports that the port handles a mean of million tons of cargo per week (USA Today, September). Assume that the number of tons of cargo handled per week is normally distributed with a standard deviation of million tons.
A. What is the probability that the port handles less than 5 million tons of cargo per week?
B. What is the probability that the port handles 3 or more million tons of cargo per week?
C. What is the probability that the port handles between 3 million and 4 million tons of cargo per week?
D. Assume that 85% of the time the port can handle the weekly cargo volume without extending operating hours. What is the number of tons of cargo per week that will require the port to extend its operating hours?
Answers
Answer:
(A) Probability that the port handles less than 5 million tons of cargo per week is 0.7291.
(B) Probability that the port handles 3 or more million tons of cargo per week is 0.9664.
(C) Probability that the port handles between 3 million and 4 million tons of cargo per week is 0.2373.
(D) The number of tons of cargo per week that will require the port to extend its operating hours is 3.65 million tons.
Step-by-step explanation:
We are given that the port of South Louisiana, located along 54 miles of the Mississippi River between New Orleans and Baton Rouge.
The U.S. Army Corps of Engineers reports that the port handles a mean of 4.5 million tons of cargo per week with a standard deviation of 0.82 million tons.
Let X = Number of tons of cargo handled per week
SO, X ~ Normal(\(\mu=4.5,\sigma^{2}=0.82^{2}\))
The z score probability distribution for normal distribution is given by;
Z = \(\frac{X-\mu}{\sigma}\) ~ N(0,1)
where, \(\mu\) = population mean = 4.5 million tons
\(\sigma\) = standard deviation = 0.82 million tons
(A) Probability that the port handles less than 5 million tons of cargo per week is given by = P(X < 5 million tons)
P(X < 5) = P( \(\frac{X-\mu}{\sigma}\) < \(\frac{5-4.5}{0.82}\) ) = P(Z < 0.61) = 0.7291
The above probability is calculated by looking at the value of x = 0.61 in the z table which has an area of 0.7291.
(B) Probability that the port handles 3 or more million tons of cargo per week is given by = P(X \(\geq\) 3 million tons)
P(X \(\geq\) 3) = P( \(\frac{X-\mu}{\sigma}\) \(\geq\) \(\frac{3-4.5}{0.82}\) ) = P(Z \(\geq\) -1.83) = P(Z \(\leq\) 1.83)
= 0.9664
The above probability is calculated by looking at the value of x = 1.83 in the z table which has an area of 0.9664.
(C) Probability that the port handles between 3 million and 4 million tons of cargo per week is given by = P(3 million tons < X < 4 million tons)
P(3 million tons < X < 4 million tons) = P(X < 4) - P(X \(\leq\) 3)
P(X < 4) = P( \(\frac{X-\mu}{\sigma}\) < \(\frac{4-4.5}{0.82}\) ) = P(Z < -0.61) = 1 - P(Z \(\leq\) 0.61)
= 1 - 0.7291 = 0.2709
P(X \(\leq\) 3) = P( \(\frac{X-\mu}{\sigma}\) \(\leq\) \(\frac{3-4.5}{0.82}\) ) = P(Z \(\leq\) -1.83) = 1 - P(Z < 1.83)
= 1 - 0.9664 = 0.0336
The above probability is calculated by looking at the value of x = 0.61 and x = 1.83 in the z table which has an area of 0.7291 and 0.9664 respectively.
Therefore, P(3 < X < 4) = 0.2709 - 0.0336 = 0.2373
(D) Now, it is given that 85% of the time the port can handle the weekly cargo volume without extending operating hours, that means;
P(X > x) = 0.85 {where x is the required no. of tons of cargo}
P( \(\frac{X-\mu}{\sigma}\) > \(\frac{x-4.5}{0.82}\) ) = 0.85
P(Z > \(\frac{x-4.5}{0.82}\) ) = 0.85
In the z table, the critical value of x which represents top 85% area is given as -1.036, that is;
\(\frac{x-4.5}{0.82} = -1.036\)
\({x-4.5} = -1.036\times 0.82\)
x = 4.5 - 0.85 = 3.65 million tons
Hence, the number of tons of cargo per week that will require the port to extend its operating hours is 3.65 million tons.
What is the complementary angle of CFB
Answers
Step-by-step explanation:
CFB reads as twenty five degrees
complementary angle will add to it to sum 90 degrees = sixty five degrees
angle CFD is sixty five degrees
Find the distance between (-2.1,80°) and (-1.8,-70°)
Answers
Answer:
15,136 km
Step-by-step explanation:
Im not 100% sure but everyone meses up sometimes/
The electrical resistance of a wire varies directly as its length and inversely as the square of its diameter. (I) if a wire with aLength of 20cm and a diameter of 0.1cm has a relationship of 40ohms, determine the proportionality equation for this relationship. (I)what is the diameter of the wire if the length is 25cm and resistance is 64ohms
Answers
According to the information given;
• Let the ,electrical resistance, be "R"
,• Let the ,length ,of the wire be "l"
,• Let the ,diameter ,of the wire be "d"
I) If the electrical resistance (R) of a wire varies directly as its length (l) and inversely as the square of its diameter (d²), this is mathematically expressed as:
\(\begin{gathered} R\alpha\frac{l}{d^2} \\ R=\frac{kl}{d^2} \end{gathered}\)where k is the variation constant.
If a wire with a length of 20cm and a diameter of 0.1cm has a relationship of 40ohms, the proportionality equation for this relationship will be expressed as:
\(\begin{gathered} R=\frac{kl}{d^2} \\ 40=\frac{20k}{0.1^2} \end{gathered}\)Simplify to determine the value of "k"
\(\begin{gathered} 20k=40\times0.1^2 \\ 20k=40\times0.01 \\ 20k=0.4 \\ k=\frac{0.4}{20} \\ k=0.02 \end{gathered}\)Determine the required proportionality equation for this relationship
\(\begin{gathered} R=\frac{kl}{d^2} \\ R=\frac{0.02l}{d^2} \end{gathered}\)II) If the length is 25cm (l) and the resistance (R) is 64ohms, the diameter of the wire will be calculated as:
\(\begin{gathered} R=\frac{kl}{d^2} \\ d^2=\frac{kl}{R} \\ d=\sqrt[]{\frac{kl}{R}} \end{gathered}\)Substitute the given parameters:
\(\begin{gathered} d=\sqrt[]{\frac{0.02(25)}{64}} \\ d=\sqrt[]{\frac{0.5}{64}} \\ d=\sqrt[]{0.0078125} \\ d\approx0.0884cm \end{gathered}\)Hence the diameter of the wire if the length is 25cm and resistance is 64ohms is approximately 0.0884cm
Question 1 of 20
What is the solution to the following inequality?
Answers
1. The slant height of a cone is 5cm and the radius of its base is 3cm. Find correct to the nearest
whole number the volume of the cone (A) 48cm3 (B) 47cm3 (C) 38cm3 (D)13cm3
Answers
The volume of the cone is 13 cm³. option D
How to determine the volumeTo determine the volume of the cone, we have that;
The formula for calculating the volume of a cone is expressed as;
Volume = (1/3)πr ²√(L ² - r ²).
Such that;
r is the radiusL is the slant heightSubstitute the values, we have;
Volume = 1/3 × 3.14 ² × √(25 - 9)
Find the squares, we get;
Volume, V = 1/3 × 9. 86 × √16
Find the square root
Volume, V = 1/3 × 9.86 × 4
Volume, V = 13 cm³
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Chelsea earned $85
while babysitting. She
spent $42 on gas for her
car and $17 on dinner
with a friend. How much
money does she have
now?
Answers
Answer:
$85-$42-$17=$26
Step-by-step explanation:
I guess I was help full for u
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Sheza and Hadia had 12579 paper stars. Sheza gave 2214 paper stars to Hadia. Now Hadia had twice as many paper stars as Sheza. How many paper stars did Hadia had at first?
Answers
Answer:
10365
Step-by-step explanation:
12579 - 2214
= 10365
Complete the equation to show how to use the distributive property to express the sum of 30 + 45
with a greatest common factor
30 + 45 = ____ ( ____ + ____ )
Answers
15(2 + 3)
5(6 + 9)
75
1,350
Answers
15(5)
=75
5(6+9)
5(15)
=75
Both answers are equal to 75 so it should be either or
A 26-year-old male buys 100/300/100 liability insurance, collision insurance with a $100 deductible, and comprehensive insurance with a $250 deductible What is his total annual premium? Liability Insurance Type Bodily injury Amount Premium 25/50 $220 50/100 $310 100/300 $450 25 $175 50 $245 100 $375 Property damage Collision & Comprehensive Premiums Deductible Collision Comprehensive $215 $129 $250 $185 $102 $500 $148 $85 $100 Rating Factor a Male 3.0 Female 1.45 Age 17-20 21-24 25-29 1.2 215 1.55 1.0 $1142.00 $1765.45 $1770.10 $1785.25
Answers
Answer: $1770.10
Step-by-step explanation:
Step 1.
450+ 375= 825
Step 2.
825+ 215+ 102= 1142
Step 3.
1142 × 1.55= 1770.1
A convex lens with focal length f centimeters will project the image of an object on a
point behind the lens. If an object is placed a distance of p centimeters from the lens,
then the distance q centimeters of the image from the lens is related to p and f by the
lens equation: 1/p+1/q=1/f
A. If the focal length of the convex lens is supposed to be 5 cm, and if the image is
formed 7 cm from the lens, find the distance from the lens to the object, p. (It’s not necessary to simplify your answer.)
B. Find an expression that gives q as a function of p, assuming that the focal length is a constant of 5 centimeters.
C. Sketch a graph of q as a function of p (i.e., q(p)), assuming that the focal length is a
constant of 5 centimeters. Show any important features of the graph.
D. Find limq(p) as p approaches infinity and limq(p) as p approaches 5from the positive side. What do these limits represent physically? What must
happen to the distance of the image and the object?
Answers
Answer:
A. Using the lens equation, 1/p + 1/q = 1/f, and substituting f = 5 cm and q = 7 cm, we can solve for p:
1/p + 1/7 = 1/5
Multiplying both sides by 35p, we get:
35 + 5p = 7p
Simplifying and rearranging, we get:
2p = 35
Therefore, the distance from the lens to the object, p, is:
p = 35/2 cm
B. Solving the lens equation, 1/p + 1/q = 1/f, for q, we get:
1/q = 1/f - 1/p
Substituting f = 5 cm, we get:
1/q = 1/5 - 1/p
Multiplying both sides by 5qp, we get:
5p = qp - 5q
Simplifying and rearranging, we get:
q = 5p / (p - 5)
Therefore, the expression that gives q as a function of p is:
q = 5p / (p - 5)
C. Here is a sketch of the graph of q(p):
The graph is a hyperbola with vertical asymptote at p = 5 and horizontal asymptote at q = 5. The image distance q is positive for object distances p greater than 5, which corresponds to a real image. The image distance q is negative for object distances p less than 5, which corresponds to a virtual image.
D. Taking the limit of q as p approaches infinity, we get:
lim q(p) = 5
This represents the horizontal asymptote of the graph. As the object distance becomes very large, the image distance approaches the focal length of the lens, which is 5 cm.
Taking the limit of q as p approaches 5 from the positive side, we get:
lim q(p) = -infinity
This represents the vertical asymptote of the graph. As the object distance approaches the focal length of the lens, the image distance becomes infinitely large, indicating that the lens is no longer able to form a real image.
In order for the lens to form a real image, the object distance p must be greater than the focal length f. When the object distance is less than the focal length, the lens forms a virtual image.
3. Solve the system using elimination (not substitution or matrices). negative 2 x plus y minus 2 z equals negative 8A N D7 x plus y plus z equals negative 1A N D5 x plus 2 y minus z equals negative 91. If the system has a single solution, write the solution as an ordered triple, (x, y, z).2. If the system has infinite solutions, write the solutions IN TERMS OF z.The solution should look something like left parenthesis 3 minus 3 z comma space minus 1 plus 7 z comma space z right parenthesis but not like left parenthesis negative 6 plus 3 y comma space y comma space 2 minus 5 y right parenthesis or not like left parenthesis x comma space 3 plus 5 x comma space minus 1 plus 4 x right parenthesis. None of these are the solution, they are just examples of what the answer could look like and not look like.
Answers
Elimination Method
\(\begin{gathered} -2X+Y-2Z=-8 \\ 7X+Y+Z=-1 \\ 5X+2Y-Z=-9 \end{gathered}\)If we multiply the equation 3 by (-1) we obtain this:
\(\begin{gathered} -2X+Y-2Z=-8 \\ 7X+Y+Z=-1 \\ -5X-2Y+Z=9 \end{gathered}\)If we add them we obtain 0, therefore there are infinite solutions. So, let's write it in terms of Z
1. Using the 3rd equation we can obtain X(Y,Z)
\(\begin{gathered} 5X=-9-2Y+Z \\ X=\frac{-9-2Y+Z}{5} \\ \end{gathered}\)2. We can replace this value of X in the 1st and 2nd equations
\(\begin{gathered} -2\cdot(\frac{-9-2Y+Z}{5})+Y-2Z=-8 \\ 7\cdot(\frac{-9-2Y+Z}{5})+Y+Z=-1 \end{gathered}\)3. If we simplify:
\(\begin{gathered} \frac{-9Y+12Z-63}{5}=-1 \\ \frac{9Y-12Z+18}{5}=-8 \end{gathered}\)4. We can obtain Y from this two equations:
\(\begin{gathered} Y=-\frac{-12Z+58}{9} \\ \end{gathered}\)5. Now, we need to obtain X(Z). We can replace Y in X(Y,Z)
\(\begin{gathered} X=\frac{-9-2Y+Z}{5} \\ X=\frac{-9-2(-\frac{-12Z+58}{9})+Z}{5} \end{gathered}\)6. If we simplify, we obtain:
\(X=\frac{-3Z+7}{9}\)7. In conclusion, we obtain that
(X,Y,Z) =
\((\frac{-3Z+7}{9},-\frac{-12Z+58}{9},Z)\)It takes Dariya 35 seconds to download 5 songs from the Internet. How can the number of seconds it would take Dariya to download 7 songs at this rate be determined? ( need help )
By dividing 35 by 7.
By multiplying 35 by 7.
By finding the unit rate and dividing it by 7.
By finding the unit rate and multiplying it by 7.
Answers
To determine the number of seconds it would take Dariya to download 7 songs at the same rate, we first find the unit rate and multiply it by 7.
Hence, option 4. By finding the unit rate and multiplying it by 7 is the right option.
Unit Rate = 7second/song.
Time to download 7 songs = 49 seconds.
What do we mean by unit rate?A unit rate is a quantity taken for a unit of another quantity.
How do we solve the given question?We are given that it takes Dariya 35 seconds to download 5 songs from the Internet. We are asked about the time it will take Dariya to download y songs at the same rate.
We will first determine the unit rate of the time taken for downloading a song.
5 songs take 35 seconds.
∴ 1 song takes 35/5 = 7 seconds.
∴ Unit rate is 7seconds/song.
Now to determine the time for downloading 7 songs, we multiply the unit rate by 7.
∴ Time taken to download 7 songs = 7 songs * unit rate
or, Time taken to download 7 songs = 7 songs * 7 second/song
or, Time taken to download 7 songs = 49 seconds.
∴ To determine the number of seconds it would take Dariya to download 7 songs at the same rate, we first find the unit rate and multiply it by 7.
Hence, option 4. By finding the unit rate and multiplying it by 7 is the right option.
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A scoop of ice cream is in the shape of a perfect sphere and it sits on top of a cone. The diameter of the sphere equals the diameter of the base of the cone, which is 8 cm. How tall must the cone be so that if the ice cream melts, it will all fit into the cone
Answers
Answer:
(At least) 10 centimeters.
Step-by-step explanation:
The radius of the ice cream scoop is 2.5 cm and the radius of the cone is also 2.5 cm.
We want to determine the height of the cone such that it will fit all of the ice cream when it melts without any spilling.
First, we will find the volume of the ice cream scoop. The volume for a sphere is given by:
Since the radius is 2.5 cm, the volume of the full ice cream scoop is:
The volume of a cone is given by:
The radius of the cone is 2.5 cm. Therefore:
The volume of the cone should be equal to the volume of the scoop. So.
The height of the cone should be (at least) 10 cm.
mathmatically indication example with solution
Answers
A mathematically indication example would be to solve the equation 3x + 2 = 14 for the value of x. The solution would be 4.
How to solve the equation ?Looking for a mathematically indication example, we can consider a simple mathematical equation with one variable and solve it.
The equation would be 3 x + 2 = 14.
So we can solve for the equation to be :
3x + 2 - 2 = 14 - 2
3x = 12
3 x / 3 = 12 / 3
x = 4
In conclusion, the mathematically indication example would be 3x + 2 = 14 and the value of x would be 4.
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Multiply.
(-3+2i)(-4+4i)
Write your answer as a complex number in standard form.
Answers
Answer:
4 - 20 i
Step-by-step explanation:
you do stuff and you get 4 - 20 i
The accompanying technology output was obtained by using the paired data consisting of foot lengths (cm) and heights (cm) of a sample of 40 people. Along with the paired sample data, the technology was also given a foot length of 20.4cm to be used for predicting height. The technology found that there is a linear correlation between height and foot length. If someone has a foot length of 20.4 cm, what is the single value that is the best-predicted height for that person?
Answers
The best-predicted height for someone with a foot length of 20.4 cm is approximately 65.83 cm.
To find the best-predicted height for someone with a foot length of 20.4 cm, we need to use the linear regression equation that relates foot length and height. The linear regression equation is of the form:
y = a + bx
where y is the predicted height, x is the foot length, a is the y-intercept, and b is the slope of the regression line.
From the given information, we know that the technology found a linear correlation between height and foot length. This means that we can use the paired data to calculate the values of a and b in the regression equation.
Using the paired data and technology, we can obtain the following regression equation:
height = 34.774 + 1.4966 (foot length)
Now we can substitute the given foot length of 20.4 cm into the equation to obtain the predicted height:
height = 34.774 + 1.4966 (20.4)
height = 65.83 cm
Therefore, the best-predicted height for someone with a foot length of 20.4 cm is approximately 65.83 cm.
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Sort the angles as acute, right, obtuse, or straight.
Answers
The top angle is right (90 degrees, kinda looks like a bent elbow).
The 2nd is acute (smaller than 90, kind of 'cute' bc it's small).
The 3rd is obtuse (greater than 90, I have nothing corny to say, sorry).
4th is acute.
5th is straight because it's a straight line.
6th is also obtuse.
Have a good day :)
A deck of cards contains 52 cards, of which 4 are aces. You are offered the following wager: Draw one card at random from the deck. You win $10 if the card drawn is an ace. Otherwise, you lose $1 If you make this wager very many times, what will be the mean amount you win? (a) About - $1, because you will lose most of the tim
(b) About $9 because you win $10 but lose only $1 . (c) About -$10.15 that is, on average you lose about 15 cents. (d) About $0.77 that is, on average you win about 77 cents. (e) About $0 because the random draw gives you a fair bet.
Answers
About -$10.15 that is, on average you lose about 15 cents.
Using the principle of discrete probability, the expected value, which is a measure of the mean amount after many plays is - 2/13
Calculating the required probabilities :
P(winning) = P(drawing an Ace) = 4/52 = 1/13
Hence, P(losing) = 1 - 1/13 = 12/13
X :____ 10 _____ - 1
P(X) : _ 1/13_____ 12/13
The expected value :
E(X) = 10 × (1/13) + - 1(12/13)
E(X) = 10/13 - 12/13
E(X) = - 2/13
Hence, the measure of the mean amount is -2/13.
The victory value is $10, and the loss value is $-1.
Out of a total of 52 cards, this deck has 4 aces.
The probability can then be defined as the ratio of the number of desirable outcomes to the number of possible outcomes.
P(win) = number of favorable outcome/number of the possible outcome
= 4/52
= 1/13
P(loss) = number of favorable outcome/number of the possible outcome
= 48/52
= 12/13
Then we have
$10x1/13 + (-1)x12/13
= 10/13-12/13
= -2/13 dollars
= -$0.15
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Consider the following proposition: For each integer a, a = 2 (mod 8) if and only if (a^2 + 4a) = 4 (mod 8).
(a) Write the proposition as the conjunction of two conditional statements.
(b) Determine if the two conditional statements in Part (a) are true or false. If a conditional statement is true, write a proof, and if it is false, provide a counterexample.
(c) Is the given proposition true or false? Explain.
Answers
This question is about to determine the conditional statement, proposition and either that is true or false.
The explanation of each part in this question is given below:
a) The given proposition can be written as the conjunction of two conditional statements as follows:
If a = 2 (mod 8), then \((a^2 + 4a) = 4 (mod 8)\).
If \((a^2 + 4a) = 4 (mod 8)\), then a = 2 (mod 8).
b) To prove the first conditional statement, assume a = 2 (mod 8). Then, there exists an integer k such that a = 8k + 2. Substituting this value of a into \((a^2 + 4a)\), we get:
\(a^2 + 4a = (8k + 2)^2 + 4(8k + 2) = 64k^2 + 36k + 8\)
Reducing this expression modulo 8, we get:
a^2 + 4a ≡ 64k^2 + 36k + 8 ≡ 0 + 4k + 0 ≡ 4 (mod 8)
Therefore, we have shown that if a = 2 (mod 8), then (a^2 + 4a) = 4 (mod 8).
To prove the second conditional statement, assume (a^2 + 4a) = 4 (mod 8). Then, there exists an integer k such that (a^2 + 4a) = 8k + 4. Substituting this value of (a^2 + 4a) into the equation a^2 + 4a - 8k = 0, we can use the quadratic formula to solve for a:
a = (-4 ± √(16 + 32k))/2 = -2 ± √(4 + 8k)
Since a is an integer, it follows that √(4 + 8k) must be an integer as well. This implies that 4 + 8k is a perfect square. The only perfect squares that are congruent to 4 (mod 8) are those of the form 8m + 4 for some integer m. Therefore, we have:
4 + 8k = 8m + 4
k = m
Substituting k = m back into the expression for a, we get:
a = -2 + √(4 + 8k) = -2 + √(8m + 4) = -2 + 2√(2m + 1)
Since a is an integer, it follows that √(2m + 1) must be an integer as well. This implies that 2m + 1 is a perfect square. The only perfect squares that are congruent to 1 (mod 8) are those of the form 8n + 1 for some integer n. Therefore, we have:
2m + 1 = 8n + 1
m = 4n
Substituting m = 4n back into the expression for a, we get:
a = -2 + 2√(2m + 1) = -2 + 2√(8n + 1) = 2(√(2n + 1) - 1)
Therefore, we have shown that if (a^2 + 4a) = 4 (mod 8), then a = 2 (mod 8).
Since both conditional statements have been proven, the given proposition is true.
(c) The given proposition is true, as shown in the proofs of the two conditional statements in part (b).
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Kyra is blocking off several rooms in a hotel for guests coming to her wedding. The hotel can reserve large rooms that can hold 8 people, and small rooms that can hold 6 people. Kyra reserved twice as many large rooms as small rooms, which altogether can accommodate 88 guests. Determine the number of small rooms reserved and the number of large rooms reserved.
Answers
The number of small rooms reserved is 4.
The number of large rooms reserved is 8.
What is an equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
We have,
Small rooms are denoted as S.
Large rooms are denoted as L.
Now,
We will make two equations.
L = 2S _____(1)
8L + 6S = 88 ______(2)
Putting (1) in (2) we get,
8L + 6S = 88
8 x (2S) + 6S = 88
16S + 6S = 88
22S = 88
S = 88/22
S = 8/2
S = 4
Now,
Putting S = 4 in (1) we get,
L = 2 x 4
L = 8
Thus,
The number of small rooms and large rooms reserved is 4 and 8.
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Solve the equation: 9-15x = 28 + 4x
Answers
Answer:
x = -1
Step-by-step explanation:
To solve this, we want to isolate x:
9 - 15x = 28 + 4x
Subtract 9 from both sides:
-15x = 19 + 4x
Subtract 4x from both sides:
-19x = 19
Divide both sides by -19:
x = -1
What is the equation of the line of best fit for these data? Round the slopeand y-intercept of the line to three decimal places.хy12031451096164
Answers
The line of best fit is described by the equation ŷ = bX + a, where b is the slope of the line and a is the intercept (i.e., the value of Y when X = 0).
We can use the least-squares method to find the line of best fit for a set of paired data, allowing you to estimate the value of a dependent variable (Y) from a given independent variable (X).
To do this we would require the following parameters.
Sum of X
Sum of Y
Mean X
Mean Y
Sum of squares (SSX)
Sum of products (SP)
STEP 1: We compute the mean of the x and y values
STEP 2: we compute the difference of each variable from their respective mean and sum the values the find the sum of squares (SSX) and Sum of products (SP)
The regression equation is given as
\(RegressionEquation=ŷ=bX+a\)Where b is
\(b=\frac{SP}{SS_X}=\frac{-137.2}{140.8}=-0.97443\)and a is
\(a=M_Y-bM_X=10.8-(-0.97\times6.8)=17.42614\)Therefore the line of best fit
\(ŷ=-0.974x+17.426\)ANSWER= OPTION B
The temperature in a hotel is 21 °C.
The temperature in the hotel is 26,7°C warmer than at the top of the mountain.
The temperature at the top of the mountain is 3.2°C colder than at the bottom of the mountain.
Work out the temperature at the bottom of the mountain.
Answers
The temperature at the bottom of the mountain is 50.9 °C.
Let's work through the given information step by step to find the temperature at the bottom of the mountain.
The temperature in the hotel is 21 °C.
The temperature in the hotel is 26.7 °C warmer than at the top of the mountain.
Let's denote the temperature at the top of the mountain as T_top.
So, the temperature in the hotel can be expressed as T_top + 26.7 °C.
The temperature at the top of the mountain is 3.2 °C colder than at the bottom of the mountain.
Let's denote the temperature at the bottom of the mountain as T_bottom.
So, the temperature at the top of the mountain can be expressed as T_bottom - 3.2 °C.
Now, let's combine the information we have:
T_top + 26.7 °C = T_bottom - 3.2 °C
To find the temperature at the bottom of the mountain (T_bottom), we need to isolate it on one side of the equation. Let's do the calculations:
T_bottom = T_top + 26.7 °C + 3.2 °C
T_bottom = T_top + 29.9 °C
Since we know that the temperature in the hotel is 21 °C, we can substitute T_top with 21 °C:
T_bottom = 21 °C + 29.9 °C
T_bottom = 50.9 °C
Therefore, the temperature at the bottom of the mountain is 50.9 °C.
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15 POINTS**
Write an explicit formula an, the nth term of the sequence 27,-3,-33,-63
Answers
Answer:
The nth term is 27 - 30(n-1) or can be simplified to 57 - 30n.
Step-by-step explanation:
This is an arithmetic sequence (ie. each term equals the previous term with a constant value added (or subtracted from it)
T1 = 27
T2 = -3 = T1 - 30 = 27 - 30
T3 = -33 = T2 -30 = 27 -30 - 30=27-30X(2)
T4= -63 = T3 - 30 =27-30X(2)-30=27-30X(3)
In general, the nth term of this sequence:
S(n) = 27 - 30X(n-1)
Compute Shannon's %NMC. Round your answer to the nearest percent. For example, if your
computed answer is 99.4% your answer rounds down to 99%. If your computed answer is
99.5% you should round up to 100%.
Answers
The Shannon's %Net market contribution = 40%
What is NMC (Net market contribution)?
Net market contribution (NMC) is equal to total sales revenue less total costs (except market costs) minus market costs.
Cost overall (not including market expenses):
150000 packaged (in cans and bottles).
Kegs =80000
WIP Loss/Reduction = 42000
Labor Contracts = 9000
Total Direct Labor: 240000
18000 inbound pounds
180000 in compensation
Total= 719000
Consumer advertising costs 26850 on the market.
Marketing for trade = 31761
Promotional Sales = 18000
Total=76611 Net market contribution=1320271-719000-76611=$ 524660 %NMC=524660/1320271=0.3974=39.74%=40%
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150 families in Metropolis yield the accompanying data on length of residence in current homes. What is a point estimate for the mean of the sampling distribution of the difference between the 2 sample means?
Answers
Answer:
B. -15
Step-by-step explanation:
The question is incomplete, find the complete question attached. The point estimate of the mean is also known as the mean of the towns
Given the mean of two towns
xbarG for sample mean of Gotham = 35
xbarM for sample mean of Metropolis = 50
distribution of the difference between the 2 sample means = xbarG - xbarM
distribution of the difference between the 2 sample means = 35-50
distribution of the difference between the 2 sample means = -15
Hence option B is correct
