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CHAPTER 18 REVIEW EXERCISES

CONCEPT CHECK EXERCISES

Determine each of the following as being either true or false. If it is false, explain why.

The ratio of 25 cm to 50 mm is 5.
If 20 m is divided into two parts in the ratio of 3/2, the parts are 14 m and 6 m.
If y varies inversely as the square of x, and $y = 4$ $y = 4$ when $x = 1/4,$ $x = 1/4,$ then $y = 4 / x^{2} .$ $y = 4 / x^{2} .$
If R varies jointly with s and the square of t, and inversely as the square of r, then $k = (\frac{R}{s}) {(\frac{r}{t})}^{2} .$ $k = (\frac{R}{s}) {(\frac{r}{t})}^{2} .$

PRACTICE AND APPLICATIONS

In Exercises 5–18, find the indicated ratios.

840 mg to 3g
300 nm to $6 μ m$ $6 μ m$
375 mL to 25 cL
12 ks to 2 h
The number $π$ $π$ equals the ratio of the circumference c of a circle to its diameter d. To check the value of $π,$ $π,$ a technician used computer simulation to measure the circumference and diameter of a metal cylinder and found the values to be $c = 4.2736 cm$ $c = 4.2736 cm$ and $d = 1.3603 cm .$ $d = 1.3603 cm .$ What value of $π$ $π$ did the technician get?
The ratio of the diagonal d of a square to the side s of the square is $\sqrt{2} .$ $\sqrt{2} .$ The diagonal and side of the face of a glass cube are found to be $d = 35.375 mm$ $d = 35.375 mm$ and $s = 25.014 mm .$ $s = 25.014 mm .$ What is the value of $\sqrt{2}$ $\sqrt{2}$ found from these measurements?
The mechanical advantage of a lever is the ratio of the output force $F_{0}$ $F_{0}$ to the input force $F_{i} .$ $F_{i} .$ Find the mechanical advantage if $F_{0} = 24 kN$ $F_{0} = 24 kN$ and $F_{i} = 5000 N .$ $F_{i} = 5000 N .$ See Fig. 18.10.

Fig. 18.10
For an automobile, the ratio of the number $n_{1}$ $n_{1}$ of teeth on the ring gear to the number $n_{2}$ $n_{2}$ of teeth on the pinion gear is the rear axle ratio of the car. Find this ratio if $n_{1} = 64$ $n_{1} = 64$ and $n_{2} = 20.$ $n_{2} = 20.$
The pressure p exerted on a surface is the ratio of the force F on the surface to its area A. Find the pressure on a square patch, 2.25 in. on a side, on a tank if the force on the patch is 42.3 lb.
The electric resistance R of a resistor is the ratio of the voltage V across the resistor to the current i in the resistor. Find R if $V = 0.632 V$ $V = 0.632 V$ and $i = 2.03 mA .$ $i = 2.03 mA .$
The heat of vaporization of a substance is the amount of heat needed to change a unit amount from liquid to vapor. Experimentation shows 7910 J are needed to change 3.50 g of water to steam. What is the heat of vaporization of water?
What is the ratio of the volume of Earth $(R = 6380 km)$ $(R = 6380 km)$ to the volume of a bacterium $(r = 0.0040 mm) ?$ $(r = 0.0040 mm) ?$
The ratio of the commission for selling a home to the selling price of the home is the commission rate. What is this rate if the commission of $17,100 is charged for selling a home priced at $380,000?
A total cholesterol level of 200 mg/dL is considered too high, but many doctors consider the ratio of the total cholesterol to HDL (high density lipids—the “good” cholesterol) a more important measure of risk to coronary heart disease. What is this ratio if the total cholesterol is 179 mg/dL and the HDL is 39 mg/dL? (Average for women is about 4.5, and for men is about 5.0.)

In Exercises 19 and 20, given that $a / b = c / d$ $a / b = c / d$ (b and d not zero), show that indicated proportions are correct.

$\frac{a + b}{b} = \frac{c + d}{d}$ $\frac{a + b}{b} = \frac{c + d}{d}$
$\frac{a - b}{b} = \frac{c - d}{d}$ $\frac{a - b}{b} = \frac{c - d}{d}$

In Exercises 21–36, answer the given questions by setting up and solving the appropriate proportions.

On a map of Hawaii, 1.3 in. represents 20.0 mi. If the distance on the map between Honolulu and Kaanapali on Maui is 6.0 in., how far is Kaanapali from Honolulu?
Given that $1.000 lb = 453.6 g,$ $1.000 lb = 453.6 g,$ what is the weight in pounds of a 14.0-g computer disk?
Given that $1.00 Btu = 1060 J,$ $1.00 Btu = 1060 J,$ how much heat in joules is produced by a heating element that produces 2660 Btu?
Given that $1.00 L = 61.0 in .^{3},$ $1.00 L = 61.0 in .^{3},$ what is the capacity in liters of a cubical box that is 3.23 in. along an edge?
A computer printer can print 60 pages in 45 s. How many pages can it print in 3 min?
A solar heater with a collector area of $58.0 m^{2}$ $58.0 m^{2}$ is required to heat 2560 kg of water. Under the same conditions, how much water can be heated by a rectangular solar collector 9.50 m by 8.75 m?
The dosage of a certain medicine is 25 mL for each 10 lb of the patient’s weight. What is the dosage for a person weighing 56 kg?
A woman invests $50,000 and a man invests $20,000 in a partnership. If profits are to be shared in the ratio that each invested in the partnership, how much does each receive from $10,500 in profits?
On a certain blueprint, a measurement of 25.0 ft is represented by 2.00 in. What is the actual distance between two points if they are 5.75 in. apart on the blueprint?
The chlorine concentration in a water supply is 0.12 part per million. How much chlorine is there in a cylindrical holding tank 4.22 m in radius and 5.82 m high filled from the water supply?
One fiber-optic cable carries 60.0% as many messages as another fiber-optic cable. Together they carry 12,000 messages. How many does each carry?
Two types of roadbed material, one 50% rock and the other 100% rock, are used in the ratio of 4 to 1 to form a roadbed. If a total of 150 tons are used, how much rock is in the roadbed?
To neutralize 80.0 kg of sodium hydroxide, 98.0 kg of sulfuric acid are needed. How much sodium hydroxide can be neutralized with 37.0 kg of sulfuric acid?
For analysis, 105 mg of DNA sample is divided into two vials in the ratio of 2 to 5. How many milligrams are in each vial?
A total of 322 bolts are in two containers. The ratio of the number of bolts in the first container to the number in the second container is 5/9. How many are in each container?
A gasoline company sells octane-87 gas and octane-91 gas in the ratio of 9 to 2. How many of each octane are sold of a total of 16.5 million gallons?

In Exercises 37–40, give the specific equation relating the variables after evaluating the constant of proportionality for the given set of values.

y varies directly as the square of x, and $y = 27$ $y = 27$ when $x = 3.$ $x = 3.$
f varies inversely as l, and $f = 500$ $f = 500$ when $l = 800.$ $l = 800.$
v is directly proportional to x and inversely proportional to the cube of y, and $v = 10$ $v = 10$ when $x = 5$ $x = 5$ and $y = 4.$ $y = 4.$
r varies jointly as u, v, and the square of w, and $r = 1/8$ $r = 1/8$ when $u = 1/2, v = 1/4,$ $u = 1/2, v = 1/4,$ and $w = 1/3.$ $w = 1/3.$

In Exercises 41–82, solve the given applied problems.

For the lever balanced at the fulcrum, the relation

$\frac{F_{1}}{F_{2}} = \frac{L_{2}}{L_{1}}$ $\frac{F_{1}}{F_{2}} = \frac{L_{2}}{L_{1}}$

holds, where $F_{1}$ $F_{1}$ and $F_{2}$ $F_{2}$ are forces on opposite sides of the fulcrum at distances $L_{1}$ $L_{1}$ and $L_{2}$ $L_{2}$ (see Fig. 18.11). Find $L_{2}$ $L_{2}$ if $F_{1} = 4.50 lb, F_{2} = 6.75 lb,$ $F_{1} = 4.50 lb, F_{2} = 6.75 lb,$ and $L_{1} = 17.5 in .$ $L_{1} = 17.5 in .$

Fig. 18.11
A company finds that the volume V of sales of a certain item and the price P of the item are related by

$\frac{P_{1}}{P_{2}} = \frac{V_{2}}{V_{1}}$ $\frac{P_{1}}{P_{2}} = \frac{V_{2}}{V_{1}}$

Find $V_{2}$ $V_{2}$ if $P_{1} = $8.00, P_{2} = $6.00,$ $P_{1} = $8.00, P_{2} = $6.00,$ and $V_{1} = 3000$ $V_{1} = 3000$ per week.
The image height h and object height H for the lens shown in Fig. 18.12 are related to the image distance q and object distance p by

$\frac{h}{H} = \frac{q}{p}$ $\frac{h}{H} = \frac{q}{p}$

Find q if

$h = 24 cm,$ $h = 24 cm,$

$H = 84 cm,$ $H = 84 cm,$ and

$p = 36 cm .$ $p = 36 cm .$

Fig. 18.12

Figure 18.12 Full Alternative Text
For two pulleys connected by a belt, the relation

$\frac{d_{1}}{d_{2}} = \frac{n_{2}}{n_{1}}$ $\frac{d_{1}}{d_{2}} = \frac{n_{2}}{n_{1}}$

holds, where d is the diameter of the pulley and n is the number of revolutions per unit time it makes. Find $n_{2}$ $n_{2}$ if $d_{1} = 4.60 in ., d_{2} = 8.30 in .,$ $d_{1} = 4.60 in ., d_{2} = 8.30 in .,$ and $n_{1} = 18.0 r / min .$ $n_{1} = 18.0 r / min .$
An apartment owner charges rent R proportional to the floor area A of the apartment. Find the equation relating R and A if an apartment of ${900 ft}^{2}$ ${900 ft}^{2}$ rents for $850/month.
The number r of aluminum cans that can be made by recycling n used cans is proportional to n. How many cans can be made from 50,000 used cans if $r = 115 cans$ $r = 115 cans$ for $n = 125 cans ?$ $n = 125 cans ?$
Under certain conditions, the rate of increase v of bacteria is proportional to the number N of bacteria present. Find v for $N = 7500$ $N = 7500$ bacteria, if $v = 800 bacteria / h$ $v = 800 bacteria / h$ for $N = 4000$ $N = 4000$ bacteria.
The index of refraction n of a medium varies inversely as the velocity of light v within it. For quartz, $n = 1.46$ $n = 1.46$ and $v = 2.05 \times 10^{8} m / s .$ $v = 2.05 \times 10^{8} m / s .$ What is n for a diamond, in which $v = 1.24 \times 10^{8} m / s ?$ $v = 1.24 \times 10^{8} m / s ?$
The force F needed to tighten a bolt varies as the length L of the wrench handle. See Fig. 18.13. If $F = 250 N$ $F = 250 N$ for $L = 22 cm,$ $L = 22 cm,$ and $F = 550 N$ $F = 550 N$ for $L = 10 cm,$ $L = 10 cm,$ determine the equation relating F and L.

Fig. 18.13
The value V of a precious stone varies directly as the square of its weight w. What is the loss in value if a stone for which $V = $27, 000$ $V = $27, 000$ is cut into two pieces with weights in the ratio of 2 to 1?
The period T of a pendulum varies directly as the square root of its length L. If $T = π / 2 s$ $T = π / 2 s$ for $L = 2.00 ft,$ $L = 2.00 ft,$ find T for $L = 4.00 ft .$ $L = 4.00 ft .$
The monthly payment p on a mortgage is directly proportional to amount A that is borrowed. If $p = $635$ $p = $635$ for $A = $100, 000.00,$ $A = $100, 000.00,$ find p if $A = $345, 000.$ $A = $345, 000.$
The velocity v (in ft/s) of water from a fire hose is directly proportional to the square root of the pressure $p (in lb / in .^{2}) .$ $p (in lb / in .^{2}) .$ If $v = 120 ft / s$ $v = 120 ft / s$ for $p = 100 lb / in .^{2},$ $p = 100 lb / in .^{2},$ find v for $p = 64 lb / in .^{2} .$ $p = 64 lb / in .^{2} .$
The component of velocity $v_{x}$ $v_{x}$ of an object moving in a circle with constant angular velocity $ω$ $ω$ varies jointly with $ω$ $ω$ and $sin ω t .$ $sin ω t .$ If $ω = π / 6 rad / s,$ $ω = π / 6 rad / s,$ and $v_{x} = - 4 π ft / s$ $v_{x} = - 4 π ft / s$ when $t = 1.00 s,$ $t = 1.00 s,$ find $v_{x}$ $v_{x}$ when $t = 9.00 s .$ $t = 9.00 s .$
The charge C on a capacitor varies directly as the voltage V across it. If the charge is $6.3 μ C$ $6.3 μ C$ with a voltage of 220 V across a capacitor, what is the charge on it with a voltage of 150 V across it?
The amount of natural gas burned is proportional to the amount of oxygen consumed. If 24.0 lb of oxygen is consumed in burning 15.0 lb of natural gas, how much air, which is 23.2% oxygen by weight, is consumed to burn 50.0 lb of natural gas?
The power P of a gas engine is proportional to the area A of the piston. If an engine with a piston area of $8.00 in .^{2}$ $8.00 in .^{2}$ can develop 30.0 hp, what power is developed by an engine with a piston area of $6.0 in .^{2} ?$ $6.0 in .^{2} ?$
The decrease in temperature at a point above a region is directly proportional to the altitude of the point above the region. If the temperature T on the ground in Toronto is $22 ° C$ $22 ° C$ and a plane 3.50 km above notes that the temperature is $1.0 ° C,$ $1.0 ° C,$ what is the temperature at the top of the CN Tower, which is 522 m high? (Assume there are no other temperature effects.)
The distance d an object falls under the influence of gravity varies directly as the square of the time t of fall. If an object falls 64.0 ft in 2.00 s, how far will it fall in 3.00 s?
The kinetic energy E of a moving object varies jointly as the mass m of the object and the square of its velocity v. If a 5.00-kg object, traveling at 10.0 m/s, has a kinetic energy of 250 J, find the kinetic energy of an 8.00-kg object moving at 50.0 m/s.
In a particular computer design, N numbers can be sorted in a time proportional to the square of log N. How many times longer does it take to sort 8000 numbers than to sort 2000 numbers?
The velocity v of a jet of fluid flowing from an opening in the side of a container is proportional to the square root of the depth d of the opening. If the velocity of the jet from an opening at a depth of 1.22 m is 4.88 m/s, what is the velocity of a jet from an opening at a depth of 7.62 m? See Fig. 18.14.

Fig. 18.14
In an electric circuit containing an inductance L and a capacitance C, the resonant frequency f is inversely proportional to the square root of the capacitance. If the resonant frequency in a circuit is 25.0 Hz and the capacitance is $95.0 μ F,$ $95.0 μ F,$ what is the resonant frequency of this circuit if the capacitance is $25.0 μ F ?$ $25.0 μ F ?$
The rate of emission R of radiant energy from the surface of a body is proportional to the fourth power of the theromodynamic temperature T. Given that a 25.0-W (the rate of emission) lamp has an operating temperature of 2500 K, what is the operating temperature of a similar 40.0-W lamp?
The frequency f of a radio wave is inversely proportional to its wavelength $λ .$ $λ .$ The constant of proportionality is the velocity of the wave, which equals the speed of light. Find this velocity if an FM radio wave has a frequency of 90.9 MHz and a wavelength of 3.29 m.
The acceleration of gravity g on a satellite in orbit around Earth varies inversely as the square of its distance r from the center of Earth. If $g = 8.7 m / s^{2}$ $g = 8.7 m / s^{2}$ for a satellite at an altitude of 400 km above the surface of Earth, find g if it is 1000 km above the surface. The radius of the Earth is $6.4 \times 10^{6} m .$ $6.4 \times 10^{6} m .$
If the weight w of an airplane varies directly as the cube of its length L, and $w = 3520 lb$ $w = 3520 lb$ for $L = 42.0 ft,$ $L = 42.0 ft,$ what would be the weight of a plane that is 50.0 ft long?
The thrust T of a propeller varies jointly as the square of the number n of revolutions per second and the fourth power of its diameter d. What is the effect on T if n is doubled and d is halved?
Using holography (a method of producing an image without using a lens), an image of concentric circles is formed. The radius r of each circle varies directly as the square root of the wavelength $λ$ $λ$ of the light used. If $r = 3.56 cm$ $r = 3.56 cm$ for $λ = 575 nm,$ $λ = 575 nm,$ find r if $λ = 483 nm .$ $λ = 483 nm .$
A metal circular ring has a circular cross section of radius r. If R is the radius of the ring (measured to the middle of the cross section), the volume V of metal in the ring varies directly as R and the square of r. If $V = {2550 mm}^{3}$ $V = {2550 mm}^{3}$ for $r = 2.32 mm$ $r = 2.32 mm$ and $R = 24.0 mm,$ $R = 24.0 mm,$ find V for $r = 3.50 mm$ $r = 3.50 mm$ and $R = 32.0 mm .$ $R = 32.0 mm .$ See Fig. 18.15.

Fig. 18.15
The range R of a stream of water from a fire hose varies jointly as the square of the initial velocity $v_{0}$ $v_{0}$ and the sine of twice the angle $θ$ $θ$ from the horizontal from which it is released. See Fig. 18.16. A stream for which $v_{0} = 35 m / s$ $v_{0} = 35 m / s$ and $θ = 22 °$ $θ = 22 °$ has a range of 85 m. Find the range if $v_{0} = 28 m / s$ $v_{0} = 28 m / s$ and $θ = 43 ° .$ $θ = 43 ° .$

Fig. 18.16
Kepler’s third law of planetary motion states that the square of the period of any planet is proportional to the cube of the mean radius (about the sun) of that planet, with the constant of proportionality being the same for all planets. Using the fact that the period of Earth is 1 year and its mean radius is 93.0 million miles, calculate the mean radius for Venus, given that its period is 7.38 months.
The stopping distance d of a car varies directly as the square of the velocity v of the car when the brakes are applied. A car moving at 32 mi/h can stop in 52 ft. What is the stopping distance for the car if it is moving at 55 mi/h?
The load L that a helical spring can support varies directly as the cube of its wire diameter d and inversely as its coil diameter D. A spring for which $d = 0.120 in .$ $d = 0.120 in .$ and $D = 0.953 in .$ $D = 0.953 in .$ can support 45.0 lb. What is the coil diameter of a similar spring that supports 78.5 lb and for which $d = 0.156 in . ?$ $d = 0.156 in . ?$
The volume rate of flow R of blood through an artery varies directly as the fourth power of the radius r of the artery and inversely as the distance d along the artery. If an operation is successful in effectively increasing the radius of an artery by 25% and decreasing its length by 2%, by how much is the volume rate of flow increased?
The safe, uniformly distributed load L on a horizontal beam, supported at both ends, varies jointly as the width w and the square of the depth d and inversely as the distance D between supports. Given that one beam has double the dimensions of another, how many times heavier is the safe load it can support than the first can support?
A bank statement exactly 30 years old is discovered. It states, “This 10-year-old account is now worth $185.03 and pays 4% interest compounded annually.” An investment with annual compound interest varies directly as $1 + r$ $1 + r$ to the power n, where r is the interest rate expressed as a decimal and n is the number of years of compounding. What was the value of the original investment, and what is it worth now?
The distance s that an object falls due to gravity varies jointly as the acceleration g due to gravity and the square of the time t of fall. The acceleration due to gravity on the moon is 0.172 of that on Earth. If a rock falls for $t_{0}$ $t_{0}$ seconds on Earth, how many times farther would the rock fall on the moon in $3 t_{0}$ $3 t_{0}$ seconds?
The heat loss L through fiberglass insulation varies directly as the time t and inversely as the thickness d of the fiberglass. If the loss through 8.0 in. of fiberglass is 1200 Btu in 30 min, what is the loss through 6.0 in. in 1 h 30 min?
A quantity important in analyzing the rotation of an object is its moment of inertia I. For a ball bearing, the moment of inertia varies directly as its mass m and the square of its radius r. Find the general expression for I if $I = 39.9 g \cdot {cm}^{2}$ $I = 39.9 g \cdot {cm}^{2}$ for $m = 63.8 g$ $m = 63.8 g$ and $r = 1.25 cm .$ $r = 1.25 cm .$
In the study of polarized light, the intensity I is proportional to the square of the cosine of the angle $θ$ $θ$ of transmission. If $I = 0.025 W / m^{2}$ $I = 0.025 W / m^{2}$ for $θ = 12.0 °,$ $θ = 12.0 °,$ find I for $θ = 20.0 ° .$ $θ = 20.0 ° .$
The force F that acts on a pendulum bob is proportional to the mass m of the bob and the sine of the angle $θ$ $θ$ the pendulum makes with the vertical. If $F = 0.120 N$ $F = 0.120 N$ for $m = 0.350 kg$ $m = 0.350 kg$ and $θ = 2.00 °,$ $θ = 2.00 °,$ find F for $m = 0.750 kg$ $m = 0.750 kg$ and $θ = 3.50 ° .$ $θ = 3.50 ° .$
A fruit-packing company plans to reduce the size of its fruit juice can (a right circular cylinder) by 10% and keep the price of each can the same (effectively raising the price). The radius and the height of the new can are to be equally proportional to those of the old can. Write one or two paragraphs explaining how to determine the percent decrease in the radius and the height of the old can that is required to make the new can.

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Table of Contents for CHAPTER 18 REVIEW EXERCISES

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Table of Contents for
CHAPTER 18 REVIEW EXERCISES