Question

The given figure shows two conical tanks and one cylindrical tank. $(\text{Volume}{)}_{\text{Cone A}}=(\text{Volume}{)}_{\text{Cone B}}=(\text{Volume}{)}_{\text{Cylinder}}=54\pi \text{cubic feet}$

$(\text{Radius}{)}_{\text{Cone A}}=(\text{Radius}{)}_{\text{Cylinder}}$

$(\text{Height}{)}_{\text{Cone B}}=(\text{Height}{)}_{\text{Cylinder}}$

If the radius $R$ and height $H$ are natural numbers, then what is the value of $h$ and$r?$

• A. $r=3\sqrt{3}\text{ft},h=18\text{ft}$
• B. $r=2\text{ft},h=12\text{ft}$
• C. $r=2\sqrt{3}\text{ft},h=12\text{ft}$
• D. $r=3\text{ft}h=18\text{ft}$

Answer 1

The correct option is A $r=3\sqrt{3}\text{ft},h=18\text{ft}$

Detailed step-by-step solution:
The radius of the cylinder = $R$
The height of the cylinder = $H$
The height of cone $A=h$
The radius of cone $A=R$
The height of cone $B=H$
The radius of cone $B=r$

Calculating the radius and height of the cylinder:
Volume of cylinder = $\pi R^{2}H$
$\Rightarrow 54\pi \text{cubic feet}=\pi R^{2}H$
$\Rightarrow 54\pi =\pi R^{2}H$
$\Rightarrow 54=R^{2}H$

Since $R\text{and}H$ are natural numbers, so $R^2$ will be a perfect square natural number.
Also, $R^2\text{and}H$ will be the factors of $54$.
Let’s check the factors of $54.$
The factors of $54\text{are}1,2,\text{and}3$.

$54=2\times 3\times 3\times 3$
$3\times 3$ will be a perfect square natural number.
So $R^2=3\times 3$
$\Rightarrow R=3\text{ft}$
And, $2\times 3$ will be a natural number.
$H=2\times 3=6\text{ft}$
Since $(\text{Volume}{)}_{\text{Cone A}}=(\text{Volume}{)}_{\text{Cylinder}}$

$\Rightarrow \frac{1}{3}\pi R^{2}h=\pi R^{2}H$

$\Rightarrow \frac{h}{3}=H$ (canceling $\pi R^2$ on both sides)

$\Rightarrow h=3H$
$\Rightarrow h=3\times 6(H=6\text{ft})$
$\Rightarrow h=18\text{ft})$

Also, $(\text{Volume}{)}_{\text{Cone B}}=(\text{Volume}{)}_{\text{Cylinder}}$

$\Rightarrow \frac{1}{3}\pi r^{2}H=\pi R^{2}H$
$\Rightarrow r^2=3R^2$ (canceling $\pi H$ on both sides)

$\Rightarrow r=\sqrt{3}R$
$\Rightarrow r=\sqrt{3}\left(3\right)(R=3\text{ft})$
$\Rightarrow r=3\sqrt{3}\text{ft})$

➡Option B is correct.