Answer:
300%
Step-by-step explanation:
Given
[tex]y\ \alpha\ \frac{1}{x^2}[/tex]
Required
Find the percentage change in y when x decreased by 50%
First, convert to equation
[tex]y\ = \ \frac{k}{x^2}[/tex]
Where k is the constant of proportionality
When x decreased by 50%
[tex]Y = \frac{k}{(x - 0.5x)^2}[/tex]
[tex]Y = \frac{k}{(0.5x)^2}[/tex]
[tex]Y = \frac{k}{0.25x^2}[/tex]
Expand
[tex]Y = \frac{1}{0.25} * \frac{k}{x^2}[/tex]
Substitute [tex]\frac{k}{x^2}[/tex] for y
[tex]Y = \frac{1}{0.25} * y[/tex]
[tex]Y = 4 * y[/tex]
[tex]Y = 4 y[/tex]
The percentage change is then calculated as:
[tex]\%Change = \frac{Y - y}{y} * 100\%[/tex]
[tex]\%Change = \frac{4y - y}{y} * 100\%[/tex]
[tex]\%Change = \frac{3y}{y} * 100\%[/tex]
[tex]\%Change = 3 * 100\%[/tex]
[tex]\%Change = 300\%[/tex]
The percentage in y is 300%
From the calculation, the percentage change in y is 300%.
From the statement of the question; y α 1/x^2 so introducing the constant of proportionality, we have; y =k/x^2 so k = x^2y.
We are then told that x is decreased by 50% hence;
Y = k/(x - 0.25x)^2
Y = k/(0.5x)^2
Y = 1/0.25 k/x^2
But k/x^2 = y so;
Y = 1/0.25 y
Y = 4y
Change = 4y -y/y * 100/1
= 300%
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