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Practice Makes Perfect
Direct and Inverse Variation
Express each of the given quantities as variations.
Total length \(L\) of cut material varies directly with the number \(n\) of identical pieces cut.
- Answer
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\(L=kn\)
For a device with constant power, current \(I\) varies inversely with voltage \(V\).
- Answer
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\(I=\frac{k}{V}\)
The area \(A\) of a geometric figure varies directly as the square of any dimension \(d\).
- Answer
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\(A=kd^2\)
For the following exercises, write an equation describing the relationship of the given variables and find the constant of proportionality for the given set of values.
\(y\) varies directly as the square of \(x\) and when \( x = 4\) , \(y = 80\) .
- Answer
- \(𝑦=k𝑥^2\), \(k=5\)
\(y\) varies inversely as the square of \(x\) and when \( x = 3\) , \(y = 2\).
- Answer
- \(𝑦=\frac{k}{x^2}\), \(k=18\)
\(y\) varies inversely as the cube root of \(x\) and when \(x = 64\) , \(y = 5\).
- Answer
- \(𝑦=\frac{k}{\sqrt[3]{x}}\), \(k=20\)
Solve Direct and Inverse Variations
In the following exercises, find the required value by setting up the variation and solving.
If \(y\) varies directly as \(x\) and \(x = 3\) when \(y = 12\). What is \(y\) when \( x = 20\).
- Answer
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80
\(a\) varies inversely as the square root of \(b\). If when \(b = 25\) , \(a = 2\), find \(a\) if \(b = 4\).
- Answer
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\(5\)
Applications of Direct and Inverse Variation
Solve
Computer Networking The time required to download a software update varies inversely with the download speed. If a file downloads in 24 minutes at 50 Mbps, how long will it take at 120 Mbps?
- Answer
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\(10 minutes\)
Electrical Technology The voltage, \(V\), across a resistor varies directly with the current, \(I\), flowing through it. A current of 3 A produces a voltage of 24 V. Write a direct variation equation and find the voltage when the current is 5 A.
- Answer
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\(V=8I\), \(40\) V
Electrical Technology The current, \(I\), drawn by a device varies inversely with the supply voltage, \(V\ when the device uses constant power. A 600-W heater draws 5 A when connected to 120 V. How much current will it draw when connected to 240 V?
- Answer
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2.5 A
Joint and Combined Variation
Express each of the given quantities as variations.
The kinetic energy \(K\) of a moving body is jointly proportional to its mass \(m\) and the square of its speed \(v\).
- Answer
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\(K=kmv^2\)
The frequency \(f\) in hertz of a guitar string is directly proportional to the speed \(v\) of sound in air and inversely proportional to the wavelength \(l\) of the sound wave.
- Answer
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\(f=\frac{kv}{l}\)
The force \(F\) exerted by the wind on a plane surface varies jointly with the square of the velocity \(v\) of the wind and with the area \(A\) of the plane surface.
- Answer
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\(F=kv^2A\)
For the following exercises, write an equation describing the relationship of the given variables and find the constant of proportionality for the given set of values.
\(x\) varies jointly as \(y\) and \(z\), and \( x = 20\) when \(y = 4\) and \(z = 2\).
- Answer
- \(x=kyz\), \(k=2.5\)
\(y\) varies jointly as \(x\) and \(z\) and inversely as \(w\). When \( x = 3\), \( z = 5\) and \( w = 6\), then \( y = 10\).
- Answer
- \(𝑦=\frac{kxz}{w}\), \(k=4\)
Solve Joint and Combination Variations
In the following exercises, find the required value by setting up the variation and solving.
\(y\) varies jointly as \(x\), \(z\) and \(w\). When \( x = 2\), \( z = 1\) and \( w = 12\) then \( y = 72\). Find \(y\) when \( x = 1\), \(z = 2\) and \( w = 3\).
- Answer
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\(y=18\)
If \(f\) varies directly as \(r\) and inversely as \(l\), and \(f=125\) when \(r=10\) and \(l=5\), then find the value of \(f\) when\(r=20\) and \(l=10\).
- Answer
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\(f=125\)
\(h\) varies jointly as the square of \(j\) and the cube of \(m\) and inversely as the square root of \(n\). When \(j = 2\) , \(m = 2\), and \(n=64\), then \(h = 12\). Find \(h\) when \(j = 1\), \(m = 3\) , and \(n = 4\).
- Answer
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\(h=\frac{81}{2}\)
Applications of Joint and Compound Variations
Solve
Physics The kinetic energy \( K\) of a moving object varies jointly with its mass \(m\) and the square of its velocity \(v\). If an object weighing 40 kilograms with a velocity of 15 meters per second has a kinetic energy of 1000 joules, find the kinetic energy if the velocity is increased to 20 meters per second.
- Answer
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\(K=\frac{16,000}{9}\approx1,777.78 joules\)
Electronics The electric power consumed by a device is directly related to the square of the voltage and inversely related to the resistance. The 18 \(\Omega\) (ohms) filament of a tube is rated at 2 W (watts) and has an operating voltage of 6 V (volts). What is the maximum voltage in a 50 \(\Omega\), 10 W resistor?
- Answer
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\(V=\sqrt500\approx22.36\) V
Physics Challenge The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute (rpm)) and the cube of the diameter. If the shaft of a certain material 3 inches in diameter can transmit 45 hp at 100 rpm, what must the diameter be in order to transmit 60 hp at 150 rpm?
- Answer
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\(d=\sqrt[3]{24}\approx2.88\) inches
Chemistry Challenge The volume of an ideal gas varies directly with the temperature and inversely with the pressure . A cylinder contains oxygen at a temperature of 310 degrees K and a pressure of 18 atmospheres in a volume of 120 liters. Find the pressure if the volume is decreased to 100 liters and the temperature is increased to 320 degrees K.
- Answer
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\(P=\frac{3,456}{155}\approx22.30\) atm
This page titled 6.2.H: Homework was last modified on Mon, 06 Jul 2026 03:20:45 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Kiersten Purves.
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