Solved Examples: Variations



Samuel Dominic Chukwuemeka (SamDom For Peace) For ACT Students
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Indicate each type of variation.
Write the proportion.
Convert to an equation.
Solve the equation.
Show all work.

(1.) (a.) The quantity, y varies directly with the square of x
If y = 24 when x = 3, determine y when x is 4.

(b.) The quantity, y varies inversely with the square of x
If y = 8 when x = 3, determine y when x is 4

(c.) The quantity, x varies directly with the square of y and inversely with z
If x = 40 when y = 4 and z = 2, determine x when y = 10 and z = 25


$ (a.) \\[3ex] y \propto x^2 ...Direct\;\;Proportion \\[3ex] y = kx^2 ...Equation...k\;\;is\;\;the\;\;proportionality\;\;constant \\[3ex] kx^2 = y \\[3ex] k = \dfrac{y}{x^2} \\[5ex] y = 24 \\[3ex] x = 3 \\[3ex] k = \dfrac{24}{3^2} \\[5ex] k = \dfrac{24}{9} \\[5ex] k = \dfrac{8}{3} \\[5ex] x = 4 \\[3ex] y = ? \\[3ex] y = kx^2 \\[3ex] y = \dfrac{8}{3} * 4^2 \\[5ex] y = \dfrac{8}{3} * 16 \\[5ex] y = \dfrac{128}{3} \\[5ex] (b.) \\[3ex] y \propto \dfrac{1}{x^2} ...Inverse\;\;Variation \\[5ex] y = \dfrac{k}{x^2} ...Equation...k\;\;is\;\;the\;\;proportionality\;\;constant \\[5ex] yx^2 = k \\[3ex] k = x^2y \\[3ex] y = 8 \\[3ex] x = 3 \\[3ex] k = 3^2 * 8 \\[3ex] k = 72 \\[3ex] x = 4 \\[3ex] y = ? \\[3ex] y = \dfrac{k}{x^2} \\[5ex] y = \dfrac{72}{4^2} \\[5ex] y = \dfrac{72}{16} \\[5ex] y = \dfrac{9}{2} \\[5ex] (c.) \\[3ex] x \propto \dfrac{y^2}{z} ...Combined\;\;Variation \\[5ex] x = \dfrac{ky^2}{z} ...Equation...k\;\;is\;\;the\;\;proportionality\;\;constant \\[5ex] \dfrac{ky^2}{z} = x \\[5ex] ky^2 = zx \\[3ex] k = \dfrac{zx}{y^2} \\[5ex] x = 40 \\[3ex] y = 4 \\[3ex] z = 2 \\[3ex] k = \dfrac{2(40)}{4^2} \\[5ex] k = \dfrac{80}{16} \\[5ex] k = 5 \\[3ex] y = 10 \\[3ex] z = 25 \\[3ex] x = ? \\[3ex] x = \dfrac{ky^2}{z} \\[5ex] x = \dfrac{5(10)^2}{25} \\[5ex] x = \dfrac{5(100)}{25} \\[5ex] x = 5(4) \\[3ex] x = 20 $
(2.) ACT Given that y varies directly as the square of x, if y = 20 when x = 2, what is y when x = 3?

$ F.\;\; 75 \\[3ex] G.\;\; 45 \\[3ex] H.\;\; 30 \\[3ex] J.\;\; 21 \\[3ex] K.\;\; 15 \\[3ex] $

$ y \propto x^2 ...Direct\;\;Proportion \\[3ex] y = kx^2 ...Equation...k\;\;is\;\;the\;\;proportionality\;\;constant \\[3ex] kx^2 = y \\[3ex] k = \dfrac{y}{x^2} \\[5ex] y = 20 \\[3ex] x = 2 \\[3ex] k = \dfrac{20}{2^2} \\[5ex] k = \dfrac{20}{4} \\[5ex] k = 5 \\[3ex] x = 3 \\[3ex] y = ? \\[3ex] y = kx^2 \\[3ex] y = 5(3)^2 \\[3ex] y = 5(9) \\[3ex] y = 45 $
(3.) ACT The table below gives the experimental data values for variables x and y.
Theory predicts that y varies directly with x.
Based on the experimental data, which of the following values is closest to the constant of variation?
(Note: The variable y varies directly with the variable x provided that y = kx for some nonzero constant k, called the constant of variation.)
x y
2.75
8.50
14.75
16.75
21.00 		0.140
0.425
0.750
0.850
1.050

$ F.\;\; -2.61 \\[3ex] G.\;\; 0.05 \\[3ex] H.\;\; 3.61 \\[3ex] J.\;\; 15.90 \\[3ex] K.\;\; 20.00 \\[3ex] $

$ y = kx \\[3ex] kx = y \\[3ex] k = \dfrac{y}{x} \\[5ex] (2.75, 0.14) \\[3ex] k = \dfrac{0.14}{2.75} \approx 0.051 \\[5ex] (8.5, 0.425) \\[3ex] k = \dfrac{0.425}{8.5} = 0.05 \\[5ex] (14.75, 0.75) \\[3ex] k = \dfrac{0.75}{14.75} \approx 0.051 \\[5ex] (16.75, 0.85) \\[3ex] k = \dfrac{0.85}{16.75} \approx 0.051 \\[5ex] (21, 1.05) \\[3ex] k = \dfrac{1.05}{21} = 0.05 \\[5ex] $ 0.05 is closest to the constant of variation.
(4.) ACT Let k be a constant of proportionality and let w, x, y, and z be positive real number variables.
In which of the following equations does x vary directly with y, directly with the square of w, and inversely with z?

$ A.\;\; x = \dfrac{kw^2}{yz} \\[5ex] B.\;\; x = \dfrac{kw^2y}{z} \\[5ex] C.\;\; x = \dfrac{ky}{w^2z} \\[5ex] D.\;\; x = \dfrac{kz}{w^2y} \\[5ex] E.\;\; x = kw^2yz \\[3ex] $

$ x \propto \dfrac{yw^2}{z} ...Combined\;\;Variation \\[5ex] x = \dfrac{kyw^2}{z} ...Equation...k\;\;is\;\;the\;\;proportionality\;\;constant \\[5ex] x = \dfrac{kw^2y}{z} $
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(10.) JAMB If x varies inversely as the cube root of y and x = 1 when y = 8, find y when x = 3

$ A.\;\; \dfrac{1}{3} \\[5ex] B.\;\; \dfrac{2}{3} \\[5ex] C.\;\; \dfrac{8}{27} \\[5ex] D.\;\; \dfrac{4}{9} \\[5ex] $

$ x \propto \dfrac{1}{\sqrt[3]{y}} ...Inverse\;\;Variation \\[5ex] x = \dfrac{k}{\sqrt[3]{y}} ...Equation...k\;\;is\;\;the\;\;proportionality\;\;constant \\[5ex] k = x \cdot \sqrt[3]{y} \\[3ex] x = 1 \\[3ex] y = 8 \\[3ex] k = 1 \cdot \sqrt[3]{8} \\[3ex] k = 1 \cdot 2 \\[3ex] k = 2 \\[3ex] x = 3 \\[3ex] y = ? \\[3ex] x \cdot \sqrt[3]{y} = k \\[3ex] \sqrt[3]{y} = \dfrac{k}{x} \\[5ex] y = \left(\dfrac{k}{x}\right)^3 \\[5ex] y = \dfrac{k^3}{x^3} \\[5ex] y = \dfrac{2^3}{3^3} \\[5ex] y = \dfrac{8}{27} $
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