Question

The height of a cone is $40cm$. A small cone is cut off at the top by a plane parallel to the base. If the volume of the small cone be $\frac{1}{64}$ of the volume of the given cone, at what height above the base is the section made ?

• A. $35cm$
• B. $30cm$
• C. $25cm$
• D. $20cm$

Answer 1

The correct option is B $30cm$

Let $r_1$ and $h$ be the radius and height of the upper cone.

Let $r_2$ be the radius of the bigger cone.

Let $v_1$ & $v_2$ be the volumes of upper cone and bigger cone respectively.

Now, $v_1=\frac{1}{64}{v}_2$

$\Rightarrow \frac{1}{3}\pi r_1^{2}h=\frac{1}{64}\times \frac{1}{3}\pi r_2^2\times 40$

$\Rightarrow {\left(\frac{r_1}{r_2}\right)}^2=\frac{40}{64}\frac{1}{h}\to \left(1\right)$.

$△ADE\sim △ABC$ thus

$\frac{AD}{AB}=\frac{DE}{BC}\Rightarrow \frac{h}{40}=\frac{r_1}{r_2}\to \left(2\right)$.

From $\left(1\right)$ and $\left(2\right)$, we get,

${\left(\frac{h}{40}\right)}^2=\frac{40}{64}\times \frac{1}{h}$

$\Rightarrow h^3=\frac{40\times 40\times 40}{4\times 4\times 4}$

$\Rightarrow h=\sqrt[3]{3\frac{40\times 40\times 40}{4\times 4\times 4}}$

$\Rightarrow h=\frac{40}{4}=10cm$.

Height of upper cone$=10cm$.

At $(40-10=30cm)$ above the base the section is made.