Question

A french fries stand at the fair serves their fries in paper cones. The cones have a radius of $1.5\text{in}$ and a height of $8\text{in}$ . They want to use new cones that have the same volume as their existing cones but a larger radius of $3\text{inches}.$
What will be the height of the new cones?

Answer 1

Detailed step-by-step solution:
The radius of each small cone, $r=1.5\text{in}$
The height of each small cone, $h=8\text{in}$
The radius of each large cone, $R=3\text{in}$
Let the height of each large cone be $H.$
Volume of a cone = $\frac{1}{3}\left(\pi r^{2}h\right)$

Volume of each small cone = Volume of each large cone
$\Rightarrow \frac{1}{3}\left(\pi r^{2}h\right)=\frac{1}{3}\left(\pi R^{2}H\right)$

$\Rightarrow r^{2}h=R^{2}H$ (canceling $\left(13\pi \right)$ term from both sides)

$\Rightarrow {1.5}^2\times 8=3^2\times H$
$1.5\times 1.5\times 8=3\times 3\times H$

$\Rightarrow H=\frac{1.5\times 1.5\times 8}{3\times 3}$

$\Rightarrow H=\frac{18}{9}$

$\Rightarrow H=2\text{inches}$
➡Option A is correct.