Deck 10: Mathematical Modeling and Variation

Question 1

The simple interest on an investment is directly proportional to the amount of the investment.By investing $2400 in a certain bond issue,you obtained an interest payment of $111.75 after 1 year.Find a mathematical model that gives the interest I for this bond issue after 1 year in terms of the amount invested P.(Round your answer to three decimal places. ) ​

A) $I = 0.047 P$
B) $I = 268,200 P$
C) $I = 21.477 P$
D) $I = 2400 P$
E) $I = 111.75 P$

Answer 1

$I = 0.047 P$

Question 2

An overhead garage door has two springs,one on each side of the door (see figure).A force of $20$ pounds is required to stretch each spring 1 foot.Because of a pulley system,the springs stretch only one-half the distance the door travels.The door moves a total of $x = 18$ feet,and the springs are at their natural length when the door is open.Find the combined lifting force applied to the door by the springs when the door is closed.​ ​

A) $$\text { Combined lifting force } = 2 F = 356 \mathrm { lb }$$
B) $\text { Combined lifting force } = 2 F = 360 \mathrm { lb }$
C) $\text { Combined lifting force } = 2 F = 362 \mathrm { lb }$
D) $\text { Combined lifting force } = 2 F = 358 \mathrm { lb }$
E) $\text { Combined lifting force } = 2 F = 364 \mathrm { lb }$

Answer 2

$\text { Combined lifting force } = 2 F = 360 \mathrm { lb }$

Question 3

The simple interest on an investment is directly proportional to the amount of the investment.By investing $5800 in a municipal bond,you obtained an interest payment of $221.25 after 1 year.Find a mathematical model that gives the interest I for this municipal bond after 1 year in terms of the amount invested P.(Round your answer to three decimal places. ) ​

A) $I = 26.215 P$
B) $I = 221.25 P$
C) $I = 0.038 P$
D) $I = 1,283,250 P$
E) $I = 5800 P$

Answer 3

$I = 0.038 P$

Question 4

Find a mathematical model representing the statement.(Determine the constant of proportionality. ) ​
Y varies inversely as x. $$( y = 9 \text { when } x = 45 . )$$ ​

A) $y = \frac { 9 } { x }$
B) $$y = \frac { 405 } { x }$$
C) $y = \frac { x } { 405 }$
D) $y = \frac { 45 } { x }$
E) $y = 405 x$

Question 5

On a yardstick with scales in inches and centimeters,you notice that 11 inches is approximately the same length as 33 centimeters.Use this information to find a mathematical model that relates centimeters y to inches x.Then use the model to find the numbers of centimeters in 60 inches and 70 inches.(Round your answer to one decimal place. ) ​

A)Model: y = $\frac { 1 } { 3 }$ x;20 cm,23.3 cm
B)Model: y = 3x;180 cm,23.3 cm
C)Model: y = 3x;20 cm,210 cm
D)Model: y = 3x;180 cm,210 cm
E)Model: y = $\frac { 1 } { 3 }$ x;180 cm,210 cm

Question 6

Find a mathematical model representing the statement.(Determine the constant of proportionality. ) ​
P varies directly as x and inversely as the square of y.(P = $\frac { 3 } { 2 }$ when x = 25 and y = 10. )
​

A) $P = \frac { x y ^ { 2 } } { 6 }$
B) $P = \frac { 6 x } { y }$
C) $P = \frac { 6 x } { y ^ { 2 } }$
D) $P = \frac { 6 y ^ { 2 } } { x }$
E) $P = 6 x y ^ { 2 }$

Question 7

Property tax is based on the assessed value of a property.A house that has an assessed value of $200,000 has a property tax of $4,820.Find a mathematical model that gives the amount of property tax y in terms of the assessed value x of the property.Use the model to find the property tax on a house that has an assessed value of $230,000.(Round your answer to four decimal places. ) ​

A) $y = 0.0241 x ; \$ 230,000$
B) $y = 0.0241 x ; \$ 5543$
C) $y = 41.4938 x ; \$ 5543$
D) $y = 41.4938 x ; \$ 9,543,568$
E) $y = 0.0241 x ; \$ 9,543,568$

Question 8

Find a mathematical model representing the statement.(Determine the constant of proportionality. ) ​
F is jointly proportional to r and the third power of s. $( F = 24750 \text { when } r = 18 \text { and } s = 5 . )$ ​

A) $F = \frac { 11 r } { s ^ { 3 } }$
B) $F = \frac { r s ^ { 3 } } { 11 }$
C) $F = 11 r s ^ { 3 }$
D) $F = \frac { 11 } { r s ^ { 3 } }$
E) $F = \frac { 11 s ^ { 3 } } { r }$

Question 9

Assume that y is directly proportional to x.Use the given x-value and y-value to find a linear model that relates y and x.​ $x = 5 , y = 24$ ​

A) $$y = - \frac { 24 } { 5 } x$$
B) $y = 24 x$
C) $y = - \frac { 5 } { 24 } x$
D) $y = \frac { 24 } { 5 } x$
E) $y = \frac { 5 } { 24 } x$

Question 10

Find a mathematical model representing the statement.(Determine the constant of proportionality. ) ​
Y is inversely proportional to x. $( y = 7 \text { when } x = 5 . )$ ​

A) $y = \frac { 5 } { x }$
B) $y = \frac { x } { 35 }$
C) $y = 35 x$
D) $y = \frac { 35 } { x }$
E) $y = \frac { 7 } { x }$

Question 11

The work W (in joules)done when lifting an object varies jointly with the mass m (in kilograms)of the object and the height h (in meters)that the object is lifted.The work done when a 120-kilogram object is lifted 1.8 meters is 2116.8 joules.How much work is done when lifting a 200-kilogram object 1.5 meters? ​

A)2960 J
B)2920 J
C)2940 J
D)2950 J
E)2930 J

Question 12

Assume that y is directly proportional to x.Use the given x-value and y-value to find a linear model that relates y and x.​ $x = 5 , y = 380$ ​

A)y = 76x
B)y = - 76x
C)y = $\frac { 1 } { 76 }$ x
D)y = - $\frac { 1 } { 76 }$ x
E)y = - 380x

Question 13

The coiled spring of a toy supports the weight of a child.The spring is compressed a distance of 1.6 inches by the weight of a 35-pound child.The toy will not work properly if its spring is compressed more than 6 inches.What is the weight of the heaviest child who should be allowed to use the toy? (Round your answer to two decimal places. ) ​

A)136.25 lb
B)126.25 lb
C)131.25 lb
D)35 lb
E)141.25 lb

Question 14

A force of 270 newtons stretches a spring 0.18 meter.What force is required to stretch the spring 0.19 meter? ​

A)295 N
B)290 N
C)285 N
D)280 N
E)270 N

Question 15

Assume that y is directly proportional to x.Use the given x-value and y-value to find a linear model that relates y and x.​ $x = 38 , y = 2400$ ​ ​

A)y = $\frac { 1200 } { 19 }$ x
B)y = - 2400x
C)y = $\frac { 19 } { 1200 }$ x
D)y = - $\frac { 19 } { 1200 }$ x​
E)y = - $\frac { 1200 } { 19 }$ x

Question 16

Find a mathematical model representing the statement.(Determine the constant of proportionality. ) ​
Z varies jointly as x and y. $( z = 128 \text { when } x = 4 \text { and } y = 8 . )$ ​

A) $z = \frac { 4 y } { x }$
B) $z = \frac { 4 } { x y }$
C) $z = \frac { x y } { 4 }$
D) $z = \frac { 4 x } { y }$
E) $z = 4 x y$

Question 17

When buying gasoline,you notice that 16 gallons of gasoline is approximately the same amount of gasoline as 51 liters.Use this information to find a linear model that relates liters y to gallons x.Then use the model to find the numbers of liters in 25 gallons and 45 gallons. (Round your answer to one decimal place. )
​

A)Model: y = $\frac { 16 } { 51 }$ x;7.8 L,14.1 L
B)Model: y = $\frac { 51 } { 16 }$ x;79.7 L,14.1 L
C)Model: y = $\frac { 51 } { 16 }$ x;7.8 L,143.4 L
D)Model: y = $\frac { 51 } { 16 }$ x;79.7 L,143.4 L
E)Model: y = $\frac { 16 } { 51 }$ x;79.7 L,143.4 L

Question 18

State sales tax is based on retail price.An item that sells for $180.99 has a sales tax of $17.4.Find a mathematical model that gives the amount of sales tax y in terms of the retail price x.Use the model to find the sales tax on a $589.99 purchase.(Round your answer to four decimal places. ) ​

A) $y = 10.4017 x ; \$ 56.72$
B) $y = 0.0961 x ; \$ 56.72$
C) $y = 0.0961 x ; \$ 589.99$
D) $y = 0.0961 x ; \$ 6,137$
E) $y = 10.4017 x ; \$ 6,137$

Question 19

A force of $f = 225$ newtons stretches a spring $s = 0.15$ meter (see figure). How far will a force of 60 newtons stretch the spring? What force is required to stretch the spring 0.1 meter? ​

A) $0.15 \mathrm {~m} ; 150 \mathrm {~N}$
B) $$0.15 \mathrm {~m} ; 225 \mathrm {~N}$$
C) $0.04 \mathrm {~m} ; 150 \mathrm {~N}$
D) $0.09 \mathrm {~m} ; 285 \mathrm {~N}$
E) $0.04 \mathrm {~m} ; 225 \mathrm {~N}$

Question 20

Assume that y is directly proportional to x.Use the given x-value and y-value to find a linear model that relates y and x.​ $$x = 2 , y = 58$$ ​

A) $y = 29 x$
B) $y = - 29 x$
C) $y = 58 x$
D) $y = - 58 x$
E) $y = 29$

Question 21

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. )
​

A) $v = \frac { 0.141 p } { q s ^ { 2 } }$ ​
B) $v = - \frac { p q } { 0.141 s ^ { 2 } }$
C) $v = \frac { 0.141 p q } { s ^ { 2 } }$
D) $v = \frac { p q } { 0.141 s ^ { 2 } }$
E) $v = - \frac { 0.141 p q } { s ^ { 2 } }$

Question 22

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. $$\begin{array}{l}\begin{array} { | c | c | c | c | c | c | } \hline x & 2 & 4 & 6 & 8 & 10 \\\hline y = \frac { k } { x ^ { 2 } } & & & & & \\\hline\end{array}\\k = 2\end{array}$$

A)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 2 & 4 & 6 & 8 & 10 \\\hline y = \frac { k } { x ^ { 2 } }& \frac { 1 } { 2 } & \frac { 1 } { 8 } & \frac { 1 } { 18 } & \frac { 1 } { 32 } & \frac { 1 } { 50 } \\\hline\end{array}$$
​
B)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 2 & 4 & 6 & 8 & 10 \\\hline y = \frac { k } { x ^ { 2 } } & \frac { 1 } {2 }&\frac { 1 } {2 } &\frac { 1 } {2 } & \frac { 1 } {2 }&\frac { 1 } {2 } \\\hline\end{array}$$
​
C)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 2 & 4 & 6 & 8 & 10 \\\hline y = \frac { k } { x ^ { 2 } } & & & & & \\& \frac { 1 } { 50 } & \frac { 1 } { 32 } & \frac { 1 } { 18 } & \frac { 1 } { 8 } & \frac { 1 } { 2 } \\\\\hline\end{array}$$

​
​
D)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 2 & 4 & 6 & 8 & 10 \\\hline y = k x ^ { 2 } & & & & & \\& \frac { 1 } { 2 } & \frac { 1 } { 8 } & \frac { 1 } { 18 } & \frac { 1 } { 8 } & \frac { 1 } { 2 } \\\\\hline\end{array}$$
​ ​
E)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 2 & 4 & 6 & 8 & 10 \\\hline y = \frac { k } { x ^ { 2 } } &2 & 4& 6& 8&10 \\\hline\end{array}$$
​

Question 23

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

A) $y = k x$ ​
B) $y = \frac { k } { x }$

Question 24

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. )
​

A)0.23 ft
B)0.58 ft
C)0.38 ft
D)0.48 ft
E)0.73 ft

Question 25

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.

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

A)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 32 & 50 & 72 & 98 & 128 \\\hline\end{array}$$
​
B)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 8 & 10 & 12 & 14 & 16 \\& & & & & \\\hline\end{array}$$
​
C)​ $$\begin{array} { | c | c | r | r | r | r | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 32 & 32 & 32 & 32 & 32 \\\hline\end{array}$$
​
​
D)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 128 & 98 & 72 & 50 & 32 \\\hline\end{array}$$
​ ​
E)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 32 & 50 & 72 & 50 & 32 \\\hline\end{array}$$
​

Question 26

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. )
​

A) $z = \frac { 3 x ^ { 2 } } { 4 y }$
B) $$z = \frac { 4 x } { 3 y }$$
C) $$z = \frac { 4 x ^ { 2 } } { 3 y }$$
D) $z = - \frac { 3 x ^ { 2 } } { 4 y }$
E) $z = - \frac { 4 x ^ { 2 } } { 3 y }$

Question 27

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. $$\begin{array} { | c | c | c | c | c | c | } \hline x & 4 & 8 & 12 & 16 & 20 \\\hline y & 1 & \frac { 1 } { 2 } & \frac { 1 } { 3 } & \frac { 1 } { 4 } & \frac { 1 } { 5 } \\\hline\end{array}$$

A) $y = 4 x$
B) $y = \frac { 1 } { x }$
C) $y = \frac { 4 } { x }$
D) $y = x$
E) $y = \frac { x } { 4 }$

Question 28

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. ​

A)373.4 vibrations / sec
B)343.4 vibrations / sec
C)353.4 vibrations / sec
D)383.4 vibrations / sec
E)363.4 vibrations / sec

Question 29

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. )
​

A)About 0.84 mi/h
B)About 0.19 mi/h
C)About -0.16 mi/h
D)About 0.49 mi/h
E)About 0.34 mi/h

Question 30

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?
​

A)About 447 ft
B)About 442 ft
C)About 432 ft
D)About 452 ft
E)About 462 ft

Question 31

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. $$\begin{array}{l}\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = \frac { k } { x ^ { 2 } } & & & & & \\\hline\end{array}\\k = 5\end{array}$$

A)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline \\y=\frac{k}{x^{2}} & \frac{5}{64} & \frac{1}{20} & \frac{5}{144} & \frac{5}{196} & \frac{5}{256}\\\\\hline\end{array}$$
​
B)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline \\y=\frac{k}{x^{2}} &\frac{5}{64} &\frac{5}{64} & \frac{5}{64} &\frac{5}{64} &\frac{5}{64} \\ \\\hline\end{array}$$
​
C)​ $\begin{array}{|c|c|c|c|c|c|}\hline x & 8 & 10 & 12 & 14 & 16 \\\hline y=\frac{k}{x^{2}} & \frac{5}{256}& \frac{5}{196}& \frac{5}{144}& \frac{1}{20}& \frac{5}{64}\\\hline\end{array}$
​
D)​ $\begin{array}{|c|c|c|c|c|c|}\hline x & 8 & 10 & 12 & 14 & 16 \\\hline y=\frac{k}{x^{2}} & \frac{5}{64}& \frac{1}{20}& \frac{5}{144}& \frac{1}{20}& \frac{5}{64}\\\hline\end{array}$
​
E)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = \frac { k } { x ^ { 2 } } & & & & & \\& 8 & 10 & 12 & 14 & 16 \\\hline\end{array}$$
​

Question 32

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.
$$\begin{array}{l}\begin{array} { | l | l | l | l | l | l | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & & & & & \\& & & & & \\\hline\end{array}\\k = \frac { 1 } { 4 }\end{array}$$

A) ​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 64 & 49 & 36 & 25 & 16 \\& & & & & \\\hline\end{array}$$
​
B)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 16 & 16 & 16 & 16 & 16 \\\hline\end{array}$$
​
C)​ $$\begin{array} { | c | r | r | r | r | r | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 16 & 25 & 36 & 25 & 16 \\\hline\end{array}$$
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D)​ $$\begin{array} { | c | r | r | r | r | r | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 16 & 25 & 36 & 49 & 64 \\\hline\end{array}$$
​ ​
E)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 8 & 10 & 12 & 14 & 16 \\& & & & & \\\hline\end{array}$$
​

Question 33

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

A) $$\begin{array} { | c | c | c | c | c | c | } \hline x & 4 & 6 & 8 & 10 & 12 \\\hline y = \frac { k } { x ^ { 2 } } & & & & & \\& \frac { 5 } { 8 } & \frac { 5 } { 18 } & \frac { 5 } { 32 } & \frac { 1 } { 10 } & \frac { 5 } { 72 } \\\hline\end{array}$$
B)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 4 & 6 & 8 & 10 & 12 \\\hline \\y = \frac { k } { x ^ { 2 } } & \frac { 5 } { 8 } & \frac { 5 } { 8 } & \frac { 5 } { 8 } & \frac { 5 } { 8 } & \frac { 5 } { 8 } \\\\\hline\end{array}$$
​
C)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 4 & 6 & 8 & 10 & 12 \\\hline \\y = \frac { k } { x ^ { 2 } } & \frac { 5 } { 72 } & \frac { 1 } { 10 } & \frac { 5 } { 32 } & \frac { 5 } { 18 } & \frac { 5 } { 8 } \\\\\hline\end{array}$$
​
​
D)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 4 & 6 & 8 & 10 & 12 \\\hline y = \frac { k } { x ^ { 2 } } & \frac { 5 } { 8 } & \frac { 5 } { 18 } & \frac { 5 } { 32 } & \frac { 5 } { 18 } & \frac { 5 } { 8 } \\\hline\end{array}$$
​
E)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 4 & 6 & 8 & 10 & 12 \\\hline y = \frac { k } { x ^ { 2 } } & 4 & 6 & 8 & 10 & 12 \\\hline\end{array}$$
​

Question 34

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.
$$\begin{array} { | c | c | c | c | c | c | } \hline x & 5 & 10 & 15 & 20 & 25 \\\hline y & 26 & 13 & \frac { 26 } { 3 } & \frac { 13 } { 2 } & \frac { 26 } { 5 } \\\hline\end{array}$$

A) $y = \frac { 130 } { x }$
B) $y = \frac { x } { 130 }$
C) $y = \frac { 1 } { x }$
D) $y = 130 x$
E) $y = x$

Question 35

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.

A)
B)
C)
D)
E)

Question 36

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. $$\begin{array} { | c | c | c | c | c | c | } \hline x & 9 & 18 & 27 & 36 & 45 \\\hline y & 2 & 4 & 6 & 8 & 10 \\\hline\end{array}$$

A) $y = \frac { 2 } { 9 } x$
B) $y = \frac { 9 } { x }$
C) $y = \frac { 2 } { 9 x }$
D) $y = \frac { 9 } { 2 x }$
E) $y = \frac { 9 } { 2 } x$

Question 37

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

A)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 256 & 196 & 144 & 100 & 64 \\\hline\end{array}$$
​
B)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 64 & 100 & 144 & 196 & 256 \\\hline\end{array}$$
​
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C)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 64 & 64 & 64 & 64 & 64 \\\hline\end{array}$$
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​
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D)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 64 & 100 & 144 & 100 & 64 \\\hline\end{array}$$
​
​
E)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 8 & 10 & 12 & 14 & 16 \\& & & & & \\\hline\end{array}$$
​

Question 38

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.
$\begin{array}{|l|l|l|l|l|l|}\hline x & 8 & 10 & 12 & 14 & 16 \\\hline y=\frac{k}{x^{2}}\\\hline\end{array}$
$k = 20$

A)​​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = \frac { k } { x ^ { 2 } } & \frac { 5 } { 16 } & \frac { 1 } { 5 } & \frac { 5 } { 36 } & \frac { 5 } { 49 } & \frac { 5 } { 64 } \\\hline\end{array}$$

B)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = \frac { k } { x ^ { 2 } } & \frac { 5 } { 16 } & \frac { 5 } { 16 } & \frac { 5 } { 16 } & \frac { 5 } { 16 } & \frac { 5 } { 16 } \\\hline\end{array}$$
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​
C)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = \frac { k } { x ^ { 2 } } & \frac { 5 } { 64 } & \frac { 5 } { 49 } & \frac { 5 } { 36 } & \frac { 1 } { 5 } & \frac { 5 } { 16 } \\\hline\end{array}$$
​​
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D)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & \frac { 5 } { 16 } & \frac { 1 } { 5 } & \frac { 5 } { 36 } & \frac { 1 } { 5 } & \frac { 5 } { 16 } \\\hline\end{array}$$
​
E)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = \frac { k } { x ^ { 2 } } & 8 & 10 & 12 & 14 & 16 \\\hline\end{array}$$
​

Question 39

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. $$\begin{array} { | c | c | c | c | c | c | } \hline x & 5 & 10 & 15 & 20 & 25 \\\hline y & - 1.5 & - 3 & - 4.5 & - 6 & - 7.5 \\\hline\end{array}$$

A) $y = - \frac { 3 } { 10 x }$
B) $y = \frac { 10 } { 3 } x$
C) $y = - \frac { 3 } { 10 } x$
D) $y = \frac { 3 } { 10 } x$
E) $y = - \frac { 10 } { 3 } x$

Question 40

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

A)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 128 & 200 & 288 & 392 & 512 \\\hline\end{array}$$
​
B)​ $$\begin{array} { | c | r | r | r | r | r | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 8 & 10 & 12 & 14 & 16 \\& & & & & \\\hline\end{array}$$
​
C)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 128 & 128 & 128 & 128 & 128 \\\hline\end{array}$$
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​
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D)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 512 & 392 & 288 & 200 & 128 \\\hline\end{array}$$
​ ​
E)​ $$\begin{array} { | c | c | c | c | c | c | } \hline x & 8 & 10 & 12 & 14 & 16 \\\hline y = k x ^ { 2 } & 128 & 200 & 288 & 200 & 128 \\\hline\end{array}$$
​

Question 41

After determining whether the variation model below is of the form $y = k x$ or $y = \frac { k } { x }$ ,find the value of k.
$$\begin{array} { | c | c | c | c | c | c | } \hline x & 20 & 40 & 60 & 80 & 100 \\\hline y & \frac { 1 } { 30 } & \frac { 1 } { 60 } & \frac { 1 } { 90 } & \frac { 1 } { 120 } & \frac { 1 } { 150 } \\\hline\end{array}$$

A) $k = \frac { 1 } { 20 }$
B) $k = \frac { 3 } { 2 }$
C) $k = \frac { 5 } { 4 }$
D) $k = \frac { 1 } { 10 }$
E) $k = \frac { 2 } { 3 }$

Question 42

The simple interest on an investment is directly proportional to the amount of the investment.By investing $6000 in a certain certificate of deposit,you obtained an interest payment of $276.00 after 1 year.Determine a mathematical model that gives the interest,I ,for this CD after 1 year in terms of the amount invested,P.

A) $I = ( 0.050 ) P$
B) $I = ( 0.041 ) P$
C) $I = ( 0.049 ) P$
D) $I = ( 0.046 ) P$
E) $I = ( 0.044 ) P$

Question 43

Find a mathematical model for the verbal statement: "m varies directly as the square of w and inversely as s."

A) $m = k \left( \frac { w } { s } \right) ^ { 2 }$
B) $m = \frac { k w ^ { 2 } } { s }$
C) $m = k w ^ { 2 } s$
D) $m = k w s ^ { 2 }$
E) $m = k \left( \frac { s } { w } \right) ^ { 2 }$

Question 44

Find a mathematical model for the verbal statement: ​
"Q is jointly proportional to the cube of h and the square root of m."
​

A) $Q = k h ^ { 3 } \sqrt { m }$
B) $Q = k \sqrt { h ^ { 3 } m }$
C) $Q = k h ^ { 2 } \sqrt [ 3 ] { m }$
D) $Q = k \sqrt [ 3 ] { h m ^ { 2 } }$
E) $Q = k \sqrt [ 3 ] { h m }$

Question 45

Hooke's law states that the magnitude of force,F,required to stretch a spring x units beyond its natural length is directly proportional to x.If a force of 3 pounds stretches a spring from its natural length of 10 inches to a length of 10.7 inches,what force will stretch the spring to a length of 11.5 inches? Round your answer to the nearest hundredth.

A) $F = 5.52$
B) $F = 6.43$
C) $F = 5.70$
D) $F = 7.29$
E) $F = 6.14$ ​

Question 46

The sales tax on an item with a retail price of $972 is $68.04.Create a variational model that gives the retail price,y,in terms of the sales tax,x,and use it to determine the retail price of an item that has a sales tax of $82.62.

A)$1182.28
B)$1151.92
C)$1180.29
D)$1192.52
E)$1124.60

Question 47

Assume that y is directly proportional to x.If x = 28 and y = 21,determine a linear model that relates y and x.

A) $y = \frac { 3 } { 5 } x$
B) $y = \frac { 4 } { 3 } x$
C) $y = \frac { 3 } { 4 } x$
D) $y = \frac { 2 } { 3 } x$
E) $y = \frac { 3 } { 2 } x$

Question 48

After determining whether the variation model below is of the form $y = k x$ or $y = \frac { k } { x }$ ,find the value of k. $$\begin{array} { | c | c | c | c | c | c | } \hline x & 154 & 161 & 168 & 175 & 182 \\\hline y & 66 & 69 & 72 & 75 & 78 \\\hline\end{array}$$

A) $k = 7$
B) $k = \frac { 7 } { 3 }$
C) $k = \frac { 1 } { 7 }$
D) $k = \frac { 3 } { 7 }$
E) $k = \frac { 7 } { 66 }$

Question 49

Determine whether the variation model below is of the form $y = k x$ or $y = \frac { k } { x }$ . x
12
24
36
48
60
Y $\frac { 1 } { 18 }$ ​ $\frac { 1 } { 36 }$ ​ $\frac { 1 } { 54 }$ ​ $\frac { 1 } { 72 }$ ​ $\frac { 1 } { 90 }$

A) $y = k x$
B) $y = \frac { k } { x }$