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Chapter 19
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Ratio, Proportion, and Variation
R4
R2 q
p
VOUT VIN
R3 R4 R1 R2
R4
1000
p
q
0.012 9
1000 R4 1000 1000
0.012
9
p
R4
1000 q
2000
1000 R4
•
•
•
501.33 R4 0.501 33 R4
•
•
501.33 R4 (1 0.501 33)
R4
•
0.001 33 0.5
1000 R4
501.33 R4(0.498 66)
R4
•
0.501 33 1000 R4
• R4
•
0.501 33(1000 R4) R4
◆◆◆
A knot is equal to 1 nautical mile
(M) per hour (1 M 1852 m).
•
501.33 0.501 33 R4 R4
•
•
•
501.33
0.498 66
1005.347 R4
CHAPTER 19 REVIEW PROBLEMS ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆
1. If y varies inversely as x, and y is 736 when x is 822, find y when x is 583.
2. If y is directly proportional to the 52 power of x, by what factor will y change when x is
tripled?
3. If y varies jointly as x and z, by what percent will y change when x is increased by 15% and
z is decreased by 4%?
4. The braking distance of an automobile varies directly as the square of the speed. If the
braking distance of a certain automobile is 11.0 m at 40.0 kmh, find the braking distance
at 90.0 kmh.
5. The rate of flow of liquid from a hole in the bottom of a tank is directly proportional to the
square root of the liquid depth. If the flow rate is 225 Lmin when the depth is 3.46 m, find
the flow rate when the depth is 1.00 m.
6. The power needed to drive a ship varies directly as the cube of the speed of the ship, and
a 77.4-kW engine will drive a certain ship at 11.2 knots (kn). Find the power needed to
propel that ship at 18.0 kn.
7. If the tensile strength of a cylindrical steel bar varies as the square of its diameter, by what
factor must the diameter be increased to triple the strength of the bar?
8. The life of an incandescent lamp varies inversely as the 12th power of the applied voltage, and the light output varies directly as the 3.5th power of the applied voltage. By what
factor will the life increase if the voltage is lowered by an amount that will decrease the
light output by 10%?
9. One of Kepler’s laws states that the time for a planet to orbit the sun varies directly as the 32
power of its distance from the sun. How many years will it take for Saturn, which is about
912 times as far from the sun as is the earth, to orbit the sun?
10. The volume of a cone varies directly as the altitude and the square of the base radius.
By what factor will the volume change if the altitude is doubled and the base radius is
halved?
11. The number of oscillations made by a pendulum in a given time is inversely proportional
to the length of the pendulum. A certain clock with a 75.00-cm-long pendulum is losing 15.00 mind. Should the pendulum be lengthened or shortened, and by how much?
12. A trucker usually makes a trip in 18.0 h at an average speed of 90.0 kmh. Find the travelling time if the speed were reduced to 75.0 kmh.
Review Problems
13. The force on the vane of a wind generator varies directly as the area of the vane and the
square of the wind velocity. By what factor must the area of a vane be increased so that the
wind force on it will be the same in a 12-kmh wind as it was in a 35-kmh wind?
14. The maximum deflection of a rectangular beam varies inversely as the product of the width
and the cube of the depth. If the deflection of a beam having a width of 15 cm and a depth of
35 cm is 7.5 mm, find the deflection if the width is made 20 cm and the depth 45 cm.
15. When an object moves at a constant speed, the travel time is inversely proportional to the
speed. If a satellite now circles the earth in 26 h 18 min, how long will it take if booster
rockets increase the speed of the satellite by 15%?
16. The life of an incandescent lamp is inversely proportional to the 12th power of the applied
voltage. If a lamp has a life of 2500 h when run at 115 V, what will its life expectancy be
when it is run at 110 V?
17. How far from the earth will a spacecraft be equally attracted by the earth and the moon?
The distance from the earth to the moon is approximately 385 000 km, and the mass of the
earth is about 82.0 times that of the moon.
18. Ohm’s law states that the electric current flowing in a circuit varies directly as the applied
voltage and inversely as the resistance. By what percent will the current change when the
voltage is increased by 25% and the resistance increased by 15%?
19. The allowable strength F of a column varies directly as its length L and inversely as its
radius of gyration r. If F 35 kg when L is 12 m and r is 3.6 cm, find F when L is 18 m
and r is 4.5 cm.
20. The time it takes a pendulum to go through one complete oscillation (the period) is directly
proportional to the square root of its length. If the period of a 1-m pendulum is 1.25 s, how
long must the pendulum be to have a period of 2.5 s?
21. If the maximum safe current in a wire is directly proportional to the 32 power of the
wire diameter, by what factor will the safe current increase when the wire diameter is
doubled?
22. Four workers take ten 6-h days to finish a job. How many workers are needed to finish a
similar job that is 3 times as large, in five 8-h days?
23. When a jet of water strikes the vane of a water turbine and is deflected through an angle ␪,
the force on the vane varies directly as the square of the jet velocity and the sine of ␪2. If
␪ is decreased from 55° to 40°, and if the jet velocity increases by 40%, by what percentage
will the force change?
24. By what factor will the kinetic energy change if the speed of a projectile is doubled and its
mass is halved? Use the fact that the kinetic energy of a moving body is directly proportional to the mass and the square of the velocity of the body.
25. The fuel consumption for a ship is directly proportional to the cube of the ship’s speed. If a
certain tanker uses 1584 U.S. gal. of diesel fuel on a certain run at 15.0 kn, how much fuel
would it use for the same run when the speed is reduced to 10.0 kn?
26. The diagonal of a certain square is doubled. By what factor does the area change?
27. A certain parachute is made from 52.0 m2 of fabric. If all of its dimensions are increased
by a factor of 1.40, how many square metres of fabric will be needed?
28. The rudder of a certain airplane has an area of 2.50 m2. By what factor must the dimensions
be scaled up to triple the area?
29. A 3.0-cm-diameter pipe fills a cistern in 5.0 h. Find the diameter of a pipe that will fill the
same cistern in 9.0 h.
30. If 315 m of fence will enclose a circular field containing 2.0 acres, what length will enclose
8.0 acres?
31. The surface area of a one-quarter model of an automobile measures 1.10 m2. Find the surface area of the full-sized car.
_
32. A certain water tank is 5.00 m high and holds 9000 L. How much would a 7.00-m-high
tank of similar shape hold?
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Chapter 19
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Ratio, Proportion, and Variation
33. A certain tractor weighs 5.80 t. If all of its dimensions were scaled up by a factor of 1.30,
what would the larger tractor be expected to weigh?
34. If a trough 2.5 m long holds 12 pailfuls of water, how many pailfuls will a similar trough
hold that is 4.0 m long?
35. A ball 4.50 in. in diameter weighs 18.0 oz. What is the weight of another ball of the same
density that is 9.00 in. in diameter?
36. A ball 4.00 inches in diameter weighs 9.00 lb. What is the weight of a ball 25.0 cm in
diameter made of the same material?
37. A worker’s contract has a cost-of-living clause that requires that his salary be proportional
to the cost-of-living index. If he earned $450 per week when the index was 9.40, how much
should he earn when the index is 12.7?
38. A woodsman pays a landowner $16.00 for each cord of wood he cuts on the landowner’s property, and he sells the wood for $150 per cord. When the price of firewood rose to $190, the landowner asked for a proportional share of the price. How much should he get per cord?
39. The series of numbers 3.5, 5.25, 7.875, . . . form a geometric progression, where the ratio of any
term to the one preceding it is a constant. This is called the common ratio. Find this ratio.
40. The specific gravity (SG) of a solid or liquid is the ratio of the density of the substance to
the density of water at a standard temperature (Eq. A45). Taking the density of water as
1.00 gcm3, find the density of copper having a specific gravity of 8.89.
Specific
Gravity
density of substance
SG density of water
A45
41. An iron casting having a surface area of 746 cm2 is heated so that all its dimensions increase
by 1.00%. Find the new area.
42. A certain boiler pipe will permit a flow of 35.5 Lmin. Buildup of scale inside the pipe
eventually reduces its diameter to three-fourths of its previous value. Assuming that
the flow rate is proportional to the cross-sectional area, find the new flow rate for the
pipe.
Writing
43. Suppose that your company makes plastic trays and is planning new ones with dimensions double those now being made. Your company president is convinced that they will
need only twice as much plastic as the older version. “Twice the size, twice the plastic,”
he proclaims, and no one is willing to challenge him. Your job is to make a presentation to
the president where you tactfully point out that he is wrong and where you explain that the
new trays will require eight times as much plastic. Write your presentation.
Team Project
44. We saw that the volumes (and hence the weights) of solids are proportional to the cube
of corresponding dimensions. Applying that idea, suppose that a sporting goods company
has designed a new line of equipment (ski packages, windsurfers, diving gear, personal
watercraft, clothing, etc.) based on the following statement:
The weights of people of similar build are proportional to the cube of their heights.
The goal of this project is to prove or disprove the given statement. Your team will use data
gathered from students on your campus in reaching a conclusion.