Solve Inverse Square Proportionality Problems: Express y in Terms of x and Find x When y=25

Question

y is inversely proportional to the square of x. A table of values for x and y is shown. x: 1, 2, 3, 4; y: 4, 1, 4/9, 1/4. a) Express y in terms of x. b) Work out the positive value of x when y = 25

Accepted Answer

a) $y=\frac{4}{x^2}$; b) $x=\frac{2}{5}$

Answer

Step 1: Recall the definition of inverse proportionality to the square of x. If y is inversely proportional to $x^2$, the general formula is $y=\frac{k}{x^2}$, where k is the constant of proportionality. Step 2: Solve for part (a): Substitute a pair of x and y values from the table into the formula to find k. Using x=1, y=4: $4=\frac{k}{1^2}$, so k=4. Substitute k=4 back into the general formula to get $y=\frac{4}{x^2}$. Verify with other pairs (e.g., x=2: $y=\frac{4}{2^2}=1$, which matches the table). Step 3: Solve for part (b): Substitute y=25 into the formula $y=\frac{4}{x^2}$. This gives $25=\frac{4}{x^2}$. Rearrange to solve for $x^2$: $x^2=\frac{4}{25}$. Take the positive square root of both sides: $x=\sqrt{\frac{4}{25}}=\frac{2}{5}$.