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An open metal bucket is in the shape of a frustum of a cone of height 21 cm with radii of its lower and upper ends as 10 cm and 20 cm respectively. Find the cost of milk which can completely fill the bucket at Rs. 40 per liter.

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Last updated date: 21st Jun 2024
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Answer
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Hint: First, find the volume of the bucket by the formula of volume of the frustum $V=\dfrac{1}{3}\pi h\left( {{r}_{1}}^{2}+{{r}_{2}}^{2}+{{r}_{1}}{{r}_{2}} \right)$. Since 1000cm$^3$ is equal to 1 liter. Use this formula to convert the volume into the capacity of the bucket. After that multiply the capacity of the bucket by the price of the milk of 1 liter which is the desired result.

Complete step by step answer:
Given: - Height of the bucket, $h=21cm$
Radii of the upper end, ${{r}_{1}}=20cm$
Radii of the lower end, ${{r}_{2}}=10cm$
Cost of milk, $C=40/l$
For the quantity of the milk, find the volume of the bucket.
The volume of the frustum is given by,
$V=\dfrac{1}{3}\pi h\left( {{r}_{1}}^{2}+{{r}_{2}}^{2}+{{r}_{1}}{{r}_{2}} \right)$
where $h$ is the height
${{r}_{1}}$ is the radius of upper end
${{r}_{2}}$ is the radius of the lower end
Substitute the values of ${{r}_{1}},{{r}_{2}}$ and $h$ in the above equation,
$\Rightarrow V=\dfrac{1}{3}\times \dfrac{22}{7}\times 21\left( {{20}^{2}}+{{10}^{2}}+20\times 10 \right)$
Solve the terms in the brackets,
$\Rightarrow V=\dfrac{1}{3}\times \dfrac{22}{7}\times 21\left( 400+100+200 \right)$
Add the terms in the brackets and cancel out the common factors from the numerator and denominator,
$\Rightarrow V=22\times 700$
Multiply the terms to get the volume of the bucket,
$\Rightarrow V=15400\text{ c}{{\text{m}}^{3}}$
As we know,
$\Rightarrow 1000\text{c}{{\text{m}}^{3}}=1l$
Then, the total capacity of the bucket is,
$\Rightarrow \dfrac{15400}{1000}=15.4l$
Since the cost of the milk is Rs. 40 per litre. The cost of 15.4 litres is,
$\Rightarrow 15.4\times 40=\text{Rs}\text{. }616$

Hence, the cost of milk which can fill the bucket at Rs. 40 per liters is Rs. 616.

Note:
The students might make the mistake of using $3.14$ instead of $\dfrac{22}{7}$ which will change the values a bit. So, take care of that. See if some particular value is mentioned, use that only.
Otherwise, it is up to you.
The frustum of the cone is the part of the cone when it is cut by a plane into two parts. The upper part of the cone remains the same in shape but the bottom part makes a frustum.
The frustum is a Latin word which means ‘piece cut off’.