The price, $P$, of an object varies directly as the square of its height, $h$. The price is $3 when the height is 20 cm. Find the price when the height is 30 cm. Answer $\underscore\underscore\underscore\underscore\underscore\underscore$ Given that $R$ varies inversely as the square of $T$, find the percentage change in $R$ when $T$ is doubled. Answer ___ %

1. Write the direct proportion relationship between price and square of height.

Since $P$ varies directly as $h^2$,
$\frac{P_2}{P_1}=\frac{h_2^2}{h_1^2}.$

2. Substitute the given values and calculate the new price.

$\frac{P_2}{3}=\frac{30^2}{20^2}$
$P_2=3\times \frac{900}{400}=3\times 2.25=6.75$

1. Write the inverse square proportional relationship.

Since $R$ varies inversely as the square of $T$,
$R \propto \frac{1}{T^2}$
so
$R=\frac{k}{T^2}$
for some constant $k$.

2. Find the new value of $R$ in terms of the original value when $T$ is doubled.

When $T$ is doubled, the new value of $T$ is $2T$.

New value of $R$:
$R_{\text{new}}=\frac{k}{(2T)^2}=\frac{k}{4T^2}=\frac{1}{4}R$

3. Calculate the percentage change in $R$.

The new value is $\frac{1}{4}$ of the original value.

Decrease:
$1-\frac{1}{4}=\frac{3}{4}$

Percentage decrease:
$\frac{3}{4}\times 100\%=75\%$