Question 1

Find a mathematical model representing the statement.(Determine the constant of proportionality. ) ​
V varies jointly as p and q and inversely as the square of s.(ν = 1.4 when p = 4.4,q = 7.3 and s = 1.8. )
​

Accepted Answer

The statement "V varies jointly as p and q and inversely as the square of s" translates to $v = k \frac{pq}{s^2}$, where $k$ is the constant of proportionality. Substituting $v = 1.4$, $p = 4.4$, $q = 7.3$, and $s = 1.8$ into the equation and solving for $k$ gives $k = 0.141$, making the correct model $v = \frac { 0.141 p q } { s ^ { 2 } }$.

Question 2

Use the given value of k to complete the table for the inverse variation model​ $y = \frac { k } { x ^ { 2 } }$ Plot the points on a rectangular coordinate system. $$\begin{array}{l}
\begin{array} { | c | c | c | c | c | c | } 
\hline x & 2 & 4 & 6 & 8 & 10 \
\hline y = \frac { k } { x ^ { 2 } } & & & & & \
\hline
\end{array}\
k = 2
\end{array}$$

Accepted Answer

The answer of Use the given value of k to...

Question 3

Determine whether the variation model below is of the form $y = k x$ or $y = \frac { k } { x }$ . x
13
26
39
52
65
Y
3
6
9
12
15

Accepted Answer

The relationship between x and y is linear, as doubling x results in doubling y, which is characteristic of a direct variation model $y = kx$.

Question 4

Use the fact that the resistance of a wire carrying an electrical current is directly proportional to its length and inversely proportional to its cross-sectional area. ​
A 10-foot piece of copper wire produces a resistance of 0.2 ohm.Use the constant of proportionality k = 0.000833 to find the diameter of the wire.
​
(Round the answer up to three decimal places. )
​

Accepted Answer

The resistance of a wire is directly proportional to its length and inversely proportional to its cross-sectional area. This means that the resistance of a wire is equal to a constant multiplied by the length of the wire divided by the cross-sectional area of the wire. The constant of proportionality is different for different materials. For copper, the constant of proportionality is 0.000833.We are given that a 10-foot piece of copper wire produces a resistance of 0.2 ohm. We can use this information to find the diameter of the wire.Let's call the diameter of the wire d. The cross-sectional area of the wire is equal to pi * (d/2)^2 = pi * d^2 / 4.The resistance of the wire is equal to 0.000833 * 10 / (pi * d^2 / 4) = 0.000833 * 40 / (pi * d^2) = 0.0104 / (pi * d^2).We are given that the resistance of the wire is 0.2 ohm. So, we can set up the following equation:0.0104 / (pi * d^2) = 0.2Solving for d, we get:d = sqrt(0.0104 / (pi * 0.2)) = 0.230 ftTherefore, the diameter of the wire is 0.230 ft, which is rounded up to three decimal places as 0.230 ft.

Question 5

Use the given value of k to complete the table for the direct variation model​ $y = k x ^ { 2 }$ . ​
Plot the points on a rectangular coordinate system.

$$\begin{array}{l}
\begin{array} { | l | l | l | l | l | l | } 
\hline x & 8 & 10 & 12 & 14 & 16 \
\hline y = k x ^ { 2 } & & & & & \
& & & & & \
\hline
\end{array}\
k = \frac { 1 } { 2 }
\end{array}$$

Accepted Answer

The answer of Use the given value of k to...

Question 6

Find a mathematical model representing the statement.(Determine the constant of proportionality. ) ​
Z varies directly as the square of x and inversely as y.(z = 36 when x = 9 and y = 3. )
​

Accepted Answer

Given that $z$ varies directly as the square of $x$ and inversely as $y$, the mathematical model can be represented as $z = k \frac{x^2}{y}$, where $k$ is the constant of proportionality. To find $k$, we use the given values: $z = 36$, $x = 9$, and $y = 3$. Substituting these values into the equation gives $36 = k \frac{9^2}{3}$. Solving for $k$ gives $k = 4$. Therefore, the correct model is $z = \frac{4x^2}{y}$, which matches choice C: $$z = \frac { 4 x ^ { 2 } } { 3 y }$$. However, the denominator should be $y$, not $3y$, indicating a typographical error in the choices provided. The correct mathematical model based on the explanation should be $z = \frac{4x^2}{y}$, which closely aligns with the structure of choice C despite the error in the denominator.

Question 7

Determine whether the variation model is of the form $y = k x$ or $y = \frac { k } { x }$ and find k.Then write a model that relates y and x. $$\begin{array} { | c | c | c | c | c | c | } 
\hline x & 4 & 8 & 12 & 16 & 20 \
\hline y & 1 & \frac { 1 } { 2 } & \frac { 1 } { 3 } & \frac { 1 } { 4 } & \frac { 1 } { 5 } \
\hline
\end{array}$$

Accepted Answer

The variation model is of the form $y = \frac { k } { x }$ because as $x$ increases, $y$ decreases proportionally. By examining the table, when $x = 4$, $y = 1$, so $k = 4$. The model that relates $y$ and $x$ is $y = \frac { 4 } { x }$.

Question 8

The frequency of vibrations of a piano string varies directly as the square root of the tension on the string and inversely as the length of the string.The middle A string has a frequency of 430 vibrations per second.Find the frequency of a string that has 1.25 times as much tension and is 1.4 times as long. ​

Accepted Answer

Let F be the frequency of the new string, T be the tension on the new string, and L be the length of the new string. Then we have:

F ∝ √T/L

We also know that:

430 = k√T/L

where k is a constant of proportionality. Solving for k, we get:

k = 430√L/T

Now we can use this value of k to find the frequency of the new string:

F = k√(1.25T)/(1.4L)
  = 430√L/T √(1.25T)/(1.4L)
  = 430/√1.4 * √(1.25)
  = 430/1.1832
  ≈ 343.4 vibrations/sec

Therefore, the answer is B.

Question 9

Use the fact that the diameter of the largest particle that can be moved by a stream varies approximately directly as the square of the velocity of the stream. ​
A stream with a velocity of $\frac { 1 } { 5 }$ mile per hour can move coarse sand particles about 0.07 inch in diameter.Approximate the velocity required to carry particles 0.2 inch in diameter.(Round your answer to two decimal places. )
​

Accepted Answer

The answer of Use the fact that the diameter of...

Question 10

Use the fact that the resistance of a wire carrying an electrical current is directly proportional to its length and inversely proportional to its cross-sectional area. ​
If #28 copper wire (which has a diameter of 0.0126 inch)has a resistance of 68.17 ohms per thousand feet,what length of #28 copper wire will produce a resistance of 30.5 ohms?
​

Accepted Answer

Given that resistance is directly proportional to length, we can use the formula $R_1/L_1 = R_2/L_2$ where $R_1 = 68.17 \, \Omega$ for $L_1 = 1000$ ft, and $R_2 = 30.5 \, \Omega$. Solving for $L_2$, we get $L_2 = (R_2 \times L_1) / R_1 = (30.5 \times 1000) / 68.17 \approx 447$ ft.

Question 11

Use the given value of k to complete the table for the inverse variation model​ $y = \frac { k } { x ^ { 2 } }$ Plot the points on a rectangular coordinate system. $$\begin{array}{l}
\begin{array} { | c | c | c | c | c | c | } 
\hline x & 8 & 10 & 12 & 14 & 16 \
\hline y = \frac { k } { x ^ { 2 } } & & & & & \
\hline
\end{array}\
k = 5
\end{array}$$

Accepted Answer

The answer of Use the given value of k to...

Question 12

Use the given value of k to complete the table for the direct variation model​ $y = k x ^ { 2 }$ . ​
Plot the points on a rectangular coordinate system.
$$\begin{array}{l}
\begin{array} { | l | l | l | l | l | l | } 
\hline x & 8 & 10 & 12 & 14 & 16 \
\hline y = k x ^ { 2 } & & & & & \
& & & & & \
\hline
\end{array}\
k = \frac { 1 } { 4 }
\end{array}$$

Accepted Answer

The answer of Use the given value of k to...

Question 13

Use the given value of k to complete the table for the inverse variation model​ $y = \frac { k } { x ^ { 2 } }$ Plot the points on a rectangular coordinate system. 
$\begin{array} { | c | c | c | c | c | c | } 
\hline x & 8 & 10 & 12 & 14 & 16 \
\hline y = \frac { k } { x ^ { 2 } } & & & & & \
\hline
\end{array}$
$k = 10$
​

Accepted Answer

The answer of Use the given value of k to...

Question 14

Determine whether the variation model is of the form $y = k x$ or $y = \frac { k } { x }$ and find k.Then write a model that relates y and x. 
$$\begin{array} { | c | c | c | c | c | c | } 
\hline x & 5 & 10 & 15 & 20 & 25 \
\hline y & 26 & 13 & \frac { 26 } { 3 } & \frac { 13 } { 2 } & \frac { 26 } { 5 } \
\hline
\end{array}$$

Accepted Answer

The variation model is of the form $y = \frac { k } { x }$ because as $x$ increases, $y$ decreases, which is characteristic of an inverse variation. To find $k$, we can use any pair of $x$ and $y$ values from the table. For example, using $x = 5$ and $y = 26$, we get $26 = \frac { k } { 5 }$, which gives $k = 130$. Therefore, the model that relates $y$ and $x$ is $y = \frac { 130 } { x }$.

Question 15

An oceanographer took readings of the water temperatures C (in degrees Celsius)at several depths d (in meters).The data collected are shown in the table. $$\begin{array} { l l } 
\text { Depth, } d & \text { Temperature, } C \
1000 & 3.8 ^ { \circ } \
2000 & 2.1 ^ { \circ } \
3000 & 1.8 ^ { \circ } \
4000 & 1.5 ^ { \circ } \
5000 & 0.5 ^ { \circ }
\end{array}$$
Sketch a scatter plot of the data.

Accepted Answer

The answer of An oceanographer took readings of the water...

Question 16

Determine whether the variation model is of the form $y = k x$ or $y = \frac { k } { x }$ and find k.Then write a model that relates y and x. $$\begin{array} { | c | c | c | c | c | c | } 
\hline x & 9 & 18 & 27 & 36 & 45 \
\hline y & 2 & 4 & 6 & 8 & 10 \
\hline
\end{array}$$

Accepted Answer

The relationship between $y$ and $x$ is direct variation, as $y$ increases linearly with $x$. The constant of variation $k$ can be found by dividing $y$ by $x$ for any pair of values, for example, $k = \frac{2}{9}$ when $x = 9$ and $y = 2$. Therefore, the model is $y = \frac{2}{9}x$.

Question 17

Use the given value of k to complete the table for the direct variation model​ $y = k x ^ { 2 }$ . ​
Plot the points on a rectangular coordinate system.
$$\begin{array}{l}
\begin{array} { | l | l | l | l | l | l | } 
\hline x & 8 & 10 & 12 & 14 & 16 \
\hline y = k x ^ { 2 } & & & & & \
& & & & & \
\hline
\end{array}\
k = 1
\end{array}$$

Accepted Answer

The answer of Use the given value of k to...

Question 18

Use the given value of k to complete the table for the inverse variation model​ $y = \frac { k } { x ^ { 2 } }$ . Plot the points on a rectangular coordinate system.
$\begin{array}{|l|l|l|l|l|l|}
\hline x & 8 & 10 & 12 & 14 & 16 \
\hline y=\frac{k}{x^{2}}\
\hline
\end{array}$
$k = 20$

Accepted Answer

The answer of Use the given value of k to...

Question 19

Determine whether the variation model is of the form $y = k x$ or $y = \frac { k } { x }$ and find k.Then write a model that relates y and x. $$\begin{array} { | c | c | c | c | c | c | } 
\hline x & 5 & 10 & 15 & 20 & 25 \
\hline y & - 1.5 & - 3 & - 4.5 & - 6 & - 7.5 \
\hline
\end{array}$$

Accepted Answer

The relationship between $x$ and $y$ is linear, as doubling $x$ also doubles $y$, indicating a direct variation ($y = kx$). The constant $k$ can be found by dividing $y$ by $x$ for any pair of values, for example, $-1.5 / 5 = -0.3$, so $k = -\frac{3}{10}$, making the model $y = -\frac{3}{10}x$.

Question 20

Use the given value of k to complete the table for the direct variation model​ $y = k x ^ { 2 }$ . ​
Plot the points on a rectangular coordinate system.
$$\begin{array}{l}
\begin{array} { | l | l | l | l | l | l | } 
\hline x & 8 & 10 & 12 & 14 & 16 \
\hline y = k x ^ { 2 } & & & & & \
\hline
\end{array}\
k = 2
\end{array}$$

Accepted Answer

The answer of Use the given value of k to...

Quiz 10: Mathematical Modeling and Variation

Find a mathematical model representing the statement.(Determine the constant of proportionality. ) V varies jointly as p and q and inversely as the square of s.(ν = 1.4 when p = 4.4,q = 7.3 and s = 1.8. )

Determine whether the variation model below is of the form $y = k x$ or $y = \frac { k } { x }$ . x 13 26 39 52 65 Y 3 6 9 12 15

Find a mathematical model representing the statement.(Determine the constant of proportionality. ) Z varies directly as the square of x and inversely as y.(z = 36 when x = 9 and y = 3. )

Determine whether the variation model is of the form $y = k x$ or $y = \frac { k } { x }$ and find k.Then write a model that relates y and x. $\begin{array} { | c | c | c | c | c | c | } \hline x & 9 & 18 & 27 & 36 & 45 \\\hline y & 2 & 4 & 6 & 8 & 10 \\\hline\end{array}$

Use the given value of k to complete the table for the inverse variation model $y = \frac { k } { x ^ { 2 } }$ . Plot the points on a rectangular coordinate system. $\begin{array}{|l|l|l|l|l|l|}\hline x & 8 & 10 & 12 & 14 & 16 \\\hline y=\frac{k}{x^{2}}\\\hline\end{array}$ $k = 20$

Determine whether the variation model is of the form $y = k x$ or $y = \frac { k } { x }$ and find k.Then write a model that relates y and x. $\begin{array} { | c | c | c | c | c | c | } \hline x & 5 & 10 & 15 & 20 & 25 \\\hline y & - 1.5 & - 3 & - 4.5 & - 6 & - 7.5 \\\hline\end{array}$

Quiz 10: Mathematical Modeling and Variation

Find a mathematical model representing the statement.(Determine the constant of proportionality. ) ​ V varies jointly as p and q and inversely as the square of s.(ν = 1.4 when p = 4.4,q = 7.3 and s = 1.8. ) ​

Determine whether the variation model below is of the form y=kxy = k xy=kx or y=kxy = \frac { k } { x }y=xk​ . x 13 26 39 52 65 Y 3 6 9 12 15

Find a mathematical model representing the statement.(Determine the constant of proportionality. ) ​ Z varies directly as the square of x and inversely as y.(z = 36 when x = 9 and y = 3. ) ​

Find a mathematical model representing the statement.(Determine the constant of proportionality. ) V varies jointly as p and q and inversely as the square of s.(ν = 1.4 when p = 4.4,q = 7.3 and s = 1.8. )

Determine whether the variation model below is of the form $y = k x$ or $y = \frac { k } { x }$ . x 13 26 39 52 65 Y 3 6 9 12 15

Find a mathematical model representing the statement.(Determine the constant of proportionality. ) Z varies directly as the square of x and inversely as y.(z = 36 when x = 9 and y = 3. )