About Solving Variation Problems:
In this section, we learned how to set up and solve a variation word problem. The idea is that we would first find a model for our given problem. From there, we would plug in the given information to find k, which is the constant of variation. Once this is done, we will update our model with the known value for k and then plug in the remaining information to solve the problem.
Test Objectives
- Demonstrate the ability to solve direct variation problems
- Demonstrate the ability to solve inverse variation problems
- Demonstrate the ability to solve combined variation problems
#1:
Instructions: Solve each variation problem.
a) The circumference of a circle varies directly with the radius. A circle with a radius of 10 inches has a circumference of approximately 62.83 inches. Find the approximate circumference of a circle when the radius is 12 inches.
b) The distance a ball rolls down an inclined plane varies directly with the square of the time in seconds it spends rolling. After 2 seconds, the ball rolls 32 feet. How far will the ball roll after 3 seconds?
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#2:
Instructions: Solve each variation problem.
a) At a local store, a cashier rang up a customer for a total of $196.02. The pre-tax amount was $180. How much will the next customer with $250 of merchandise pay after taxes?
b) Under ideal conditions, the power obtained from a local windmill varies directly with the cube of the wind speed. Earlier today when conditions were ideal, the windmill was producing 768 watts of power when the wind was blowing at 40 miles per hour. How much power in watts will the windmill produce under ideal conditions when the wind is blowing at 20 miles per hour?
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#3:
Instructions: Solve each variation problem.
a) The speed of a pulley varies inversely with its diameter. At a local workshop, a pulley with a diameter of 4 inches has a speed of 112.5 revolutions per minute. Find the speed of a similar pulley with a diameter of 12 inches.
b) The amount of time it takes to go a given distance varies inversely with the rate of speed. On a field trip, a bus drives from New Orleans, LA to Orlando, FL for 13 hours at an average speed of 50 miles per hour. How long will the same trip take a teacher who averages 65 miles per hour?
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#4:
Instructions: Solve each variation problem.
a) The electrical resistance of a wire varies directly with its length and inversely with the square of its diameter. A wire that is 4 meters long and 0.007 meters in diameter has a resistance of approximately 238 ohms. Find the approximate resistance in ohms of a wire made of the same material that is 5 meters long and 0.005 meters in diameter.
b) The intensity of sound varies inversely with the square of the distance from the source of the sound. Jennifer is at a concert and decides to move to a seat that is double the distance from the loudspeakers. How would the intensity of the sound at her new seat compare to the intensity of the sound at her previous seat?
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#5:
Instructions: Solve each variation problem.
a) The apparent brightness of a car headlight varies directly with the luminosity (measured in watts) and varies inversely with the square of the distance (measured in meters) from the headlight. If the luminosity was quadrupled (multiplied by 4) and the distance from the headlight was 1/3 of its original, by what factor will the brightness change?
b) The owners of a local gym find that the sales of memberships each month vary directly with their advertising budget and inversely with the monthly membership fee. If 2000 new members are added when the advertising budget is $50,000 and the price of their monthly membership is $50, how many new members are added when the advertising budget is $75,000 and the price of their monthly membership is $60?
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Written Solutions:
#1:
Solutions:
a) 75.4 inches
b) 72 feet
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#2:
Solutions:
a) $272.25
b) 96 watts
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#3:
Solutions:
a) 37.5 revolutions per minute
b) 10 hours
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#4:
Solutions:
a) 583.1 ohms
b) 0.25
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#5:
Solutions:
a) 36
b) 2500
