Question

A bucket is in the form of frustum of cone and holds $28.90$ litres of water. The radii of the top and bottom are $28$ cm and $21$ cm respectively. Find the height of the bucket.

Answer 1

Hint: Here we will proceed by using the formula for frustum of cone that is $V = \dfrac{1}{3} \times \pi \times h \times \left( {{r_1}^2 + {r_2}^2 + \left( {{r_{1 \times }}{r_2}} \right)} \right)$. Then by applying the conditions given in the question we will get our answer.

Complete step-by-step solution -

It is given that,
Volume of bucket:
=$28.490$litres

$\because {1}{m^3} =1000 litre $
 $\therefore {100 cm \times 100 cm \times 100 cm} = 1000 litre $
$\therefore { 1 {{cm}^3} } = {10}^{(-3)} litre$

So, 1 litre =$1000{{cm}^{3}} $

Hence volume of bucket = $28.490$ litres = $28490 c{m^3}$
Let the height of the bucket be $h$ cm
We have:
$r = 21$ cm,
And $R = 28$cm
Therefore, $\dfrac{\pi }{3}h\left[ {{{\left( {28} \right)}^2} + {{\left( {21} \right)}^2} + 28 \times 21} \right] = 28490$
$ \Rightarrow h\left( {784 + 441 + 588} \right) = \dfrac{{28490 \times 21}}{{22}}$
By simplifying, we will get,
$ \Rightarrow 1813h = 27195$
$
   \Rightarrow h = \dfrac{{27195}}{{1813}} \\
   \Rightarrow h = 15cm \\
 $
Therefore, height of bucket $ = 15cm$

Note: Whenever we come up with this type of question, one must know that frustum may be formed from a right circular cone by cutting off the tip of the cone with a cut perpendicular to the height, forming a lower base and an upper base that are circular and parallel. In this way one can easily solve these questions.