Question 1

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

Accepted Answer

Let the length of the wire be x feet.
According to the given information,
Resistance of 1000 ft of #28 copper wire = 68.17 ohms
Resistance of 1 ft of #28 copper wire = 0.06817 ohms
Diameter of #28 copper wire = 0.0126 inch
Radius of #28 copper wire = 0.0063 inch
Area of cross section of #28 copper wire = π(0.0063)^2 = 0.0001247 sq. inch
Resistance of x ft of #28 copper wire = (0.06817/1000) * x ohms
Also, from the given information,
Resistance is directly proportional to length.
So, R1/R2 = L1/L2
or, 68.17/((0.06817/1000)x) = 1000/ x
or, x = (1000 * (0.06817/1000)) / (68.17/30.5) = 447.01 feet
Therefore, the length of #28 copper wire required to produce a resistance of 30.5 ohms is about 447 feet.

Question 2

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) $z = - \frac { 4 x ^ { 2 } } { 3 y }$

Accepted Answer

Given that $z$ varies directly as the square of $x$ and inversely as $y$, the model is $z = k \frac{x^2}{y}$, where $k$ is the constant of proportionality. Substituting $x = 9$, $y = 3$, and $z = 36$ into the model gives $36 = k \frac{81}{3}$, solving for $k$ gives $k = 4/3$. Thus, the correct model is $z = \frac{4x^2}{3y}$.

Question 3

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) $k = \frac { 7 } { 66 }$

Accepted Answer

The answer of After determining whether the variation model below...

Question 4

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

Accepted Answer

Let's define the sales tax rate as $r$. Then, we know that:

$r = \dfrac{68.04}{972} \approx 0.07$

This means that the sales tax rate is approximately 7%.

We can set up a variational model as:

$y = (1+r)x$

where $y$ is the retail price, $x$ is the sales tax, and $r$ is the sales tax rate.

Plugging in the values we know, we get:

$y = (1+0.07)(82.62) \approx 1180.29$

Therefore, the retail price of an item with a sales tax of $82.62 is approximately $1180.29.

The best choice is C.

Question 5

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) $I = ( 0.044 ) P$

Accepted Answer

The interest rate can be found by dividing the interest payment by the principal amount: $ \frac{276}{6000} = 0.046 $. Thus, the model is $ I = (0.046) P $.

Question 6

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

Accepted Answer

The frequency $f$ of a string is given by $f \propto \frac{\sqrt{T}}{L}$, where $T$ is the tension and $L$ is the length. If the tension is increased by a factor of 1.25, the square root of the tension increases by a factor of $\sqrt{1.25}$. If the length is increased by a factor of 1.4, the frequency becomes $430 \times \frac{\sqrt{1.25}}{1.4}$. Calculating this gives a frequency of approximately 343.4 vibrations per second.

Question 7

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

Accepted Answer

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

Question 8

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) $m = k \left( \frac { s } { w } \right) ^ { 2 }$

Accepted Answer

The correct mathematical model for "m varies directly as the square of w and inversely as s" is $m = \frac { k w ^ { 2 } } { s }$, where $k$ is the constant of proportionality. This captures both the direct variation with the square of $w$ and the inverse variation with $s$.

Question 9

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

Accepted Answer

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

Question 10

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) $y = \frac { 3 } { 2 } x$

Accepted Answer

Since y is directly proportional to x, y = kx. Given x = 28 and y = 21, k = 21/28 = 3/4. Thus, the model is y = (3/4)x.

Question 11

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) $Q = k \sqrt [ 3 ] { h m }$

Accepted Answer

The statement "Q is jointly proportional to the cube of h and the square root of m" translates to $Q = k h^3 \sqrt{m}$, where $k$ is the constant of proportionality.

Question 12

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) $F = 6.14$

Accepted Answer

Hooke's law is given by $F = kx$, where $F$ is the force applied, $x$ is the extension from the natural length, and $k$ is the spring constant. Given that a force of 3 pounds stretches the spring 0.7 inches (from 10 to 10.7 inches), we can find $k$ by rearranging the formula to $k = \frac{F}{x} = \frac{3}{0.7}$. To find the force required to stretch the spring to 11.5 inches (1.5 inches extension), we use $F = kx = \frac{3}{0.7} \times 1.5$, which gives approximately 6.43 pounds.

Question 13

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.(v = 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) $v = - \frac { 0.141 p q } { s ^ { 2 } }$

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. To find $k$, substitute $v = 1.4$, $p = 4.4$, $q = 7.3$, and $s = 1.8$ into the equation: $1.4 = k \frac{4.4 \times 7.3}{1.8^2}$. Solving for $k$ gives $k = 0.141$, so the correct model is $v = \frac{0.141 pq}{s^2}$.

Quiz 1: Functions and Their Graphs

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.)

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

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.

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

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

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

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.(v = 1.4 when p = 4.4, q = 7.3 and s = 1.8.)