H varies directly as p

H ∝ p

H varies inversely as the square of y

H ∝ $\frac{1}{{\mathrm{y}}^2}$

Combining the two

H ∝ $\frac{\mathrm{p}}{{\mathrm{y}}^2}$

Introduce a constant k to remove the proportionality sign.

H = k x $\frac{\mathrm{p}}{{\mathrm{y}}^2}$

Use the given values to find k.

H = 1

p = 8

y = 2

1 = k x $\frac{8}{2^2}$

1 = k x $\frac{8}{4}$

1 = k x 2

Divide both sides by 2

k = $\frac{1}{2}$

Substitute the value of k into the general relation.

H = k x $\frac{\mathrm{p}}{{\mathrm{y}}^2}$

H = $\frac{1}{2}$ x $\frac{\mathrm{p}}{{\mathrm{y}}^2}$

H = $\frac{\mathrm{1\; x\; p}}{\mathrm{2\; x}{\mathrm{y}}^2}$

H = $\frac{\mathrm{p}}{2{\mathrm{y}}^2}$

y ∝ $\frac{1}{{\mathrm{x}}^2}$

NB: ∝ is the symbol for proportional
inversely proportional is 1 over the expression

Introduce a constant to remove the proportionality sign.

y = k x $\frac{1}{{\mathrm{x}}^2}$

Use the known values to calculate the constant(k) to get the general expression

y = 100 and x = 3

100 = k x $\frac{1}{3^2}$

100 = k x $\frac{1}{9}$

Get rid of the fraction by multiplying both sides of the equation by the denominator 9

100 x 9 = k x $\frac{1}{9}$ x 9

100 x 9 = k x 1

k = 900

Substitute the value of k to get the general equation.

y = 900 x $\frac{1}{{\mathrm{x}}^2}$ = $\frac{900}{{\mathrm{x}}^2}$

When y = 25

25 = $\frac{900}{{\mathrm{x}}^2}$

Get rid of the fraction by multiplying both sides of the equation by the denominator x²

25 x x² = $\frac{900}{{\mathrm{x}}^2}$ x x²

25 x x² = 900

Divide both sides by 25

x² = $\frac{900}{25}$

x² = 36

Take square root of both sides to get rid of the square.

x = $\sqrt{\mathrm{36}}$ = 6

y varies inversely as x ⇒ y ∝ $\frac{1}{\mathrm{x}}$

Remove the proportionality sign (∝) by introducing a constant (k)

y = k x $\frac{1}{\mathrm{x}}$ = $\frac{\mathrm{k}}{\mathrm{x}}$

When x = 3, y = 6.

6 = $\frac{\mathrm{k}}{3}$

Solve for k.

k = 6 x 3 = 18

y = $\frac{18}{\mathrm{x}}$

When x = 9

y = $\frac{18}{9}$ = 2