# What Height Above the Base is the Section Made? If

The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be 1/27th of the value of the given cone, at what height above the base is the section made?

- 19 cm
- 20 cm
- 12 cm
- 15 cm

Answer: (2) 20 cm

Explanation:-

Let the height and radius of big cone be h_{1} and r_{1.}

and the height and radius of cut off small cone be h and r.

We know that, from similar triangle,

h_{1} / r_{1} = h_{2} / r_{2}

h_{2} = h_{1} r_{2} / r_{1}

Given- h_{1} = 30

h_{2} = 30 (r_{2} /r_{1}) ——–(1)

Volume of big cone, V_{1} = (1/3) π (r_{1}^{2}) h_{1}

volume of small cone,V_{2}= (1/3) π r^{2} h_{2}

V_{1} / V_{2} = (r_{1}^{2}) h_{1}/ (r_{2}^{2} h = 27 (Given- the volume of small cone is 1/27 of big cone)

(r) h_{1} = 27(r_{2}^{2}) h_{2}

30 (r_{1}^{2}) = 27(r_{2}^{2})h_{2}

**
**h

_{1}= 30(r

_{1}

^{2}) / 27 (r

_{2}

^{2})

h_{2} = (30 /27)(r_{1} /r_{2})^{2} ———(2)

From equating (1) and (2)

30 (r_{2} /r_{1}) = (30 /27) (r_{1}/r_{2})^{2}

(r_{2} /r_{1})^{3} = 1 / 27

r_{2} / r_{1} = 1/3

substituting this in (1) h_{2} = 30 (r_{2} /r_{1})

h_{2 }= 30 (1/3) = 10 cm

the height of section made is 30 – 10 = 20 cm.

Hence, the answer is (2) 20 cm.