$A$ container,opened from the top and made up of a metal sheet,is in the form of a frustum of a cone of height $16 \, cm$ with radii of its lower and upper ends as $8 \, cm$ and $20 \, cm$,respectively. Find the cost of the milk which can completely fill the container,at the rate of $Rs. \, 20$ per litre. Also,find the cost of the metal sheet used to make the container,if it costs $Rs. \, 8$ per $100 \, cm^2$. (Take $\pi = 3.14$)

Vedclass pdf generator app on play store
Vedclass iOS app on app store
(N/A) Radius $(r_1)$ of upper end of container $= 20 \, cm$
Radius $(r_2)$ of lower end of container $= 8 \, cm$
Height $(h)$ of container $= 16 \, cm$
Slant height $(l)$ of frustum $= \sqrt{(r_1 - r_2)^2 + h^2} = \sqrt{(20 - 8)^2 + 16^2} = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \, cm$
Capacity of container $=$ Volume of frustum $= \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 r_2)$
$= \frac{1}{3} \times 3.14 \times 16 \times (20^2 + 8^2 + 20 \times 8) = \frac{1}{3} \times 3.14 \times 16 \times (400 + 64 + 160) = \frac{1}{3} \times 3.14 \times 16 \times 624 = 10449.92 \, cm^3 = 10.44992 \, litres \approx 10.45 \, litres$
Cost of $1 \, litre$ milk $= Rs. \, 20$
Cost of $10.45 \, litres$ milk $= 10.45 \times 20 = Rs. \, 209$
Area of metal sheet used $= \pi (r_1 + r_2) l + \pi r_2^2 = 3.14 \times (20 + 8) \times 20 + 3.14 \times 8^2 = 3.14 \times 28 \times 20 + 3.14 \times 64 = 1758.4 + 200.96 = 1959.36 \, cm^2$
Cost of $100 \, cm^2$ metal sheet $= Rs. \, 8$
Cost of $1959.36 \, cm^2$ metal sheet $= \frac{1959.36 \times 8}{100} = Rs. \, 156.7488 \approx Rs. \, 156.75$
Therefore,the cost of the milk is $Rs. \, 209$ and the cost of the metal sheet is $Rs. \, 156.75$.

Explore More

Similar Questions

$A$ copper rod of diameter $1 \,cm$ and length $8 \,cm$ is drawn into a wire of length $18 \,m$ of uniform thickness. Find the thickness of the wire. (in $mm$)

$A$ tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are $2.1 \, m$ and $4 \, m$ respectively,and the slant height of the top is $2.8 \, m$,find the area of the canvas used for making the tent. Also,find the cost of the canvas of the tent at the rate of $Rs. \, 500$ per $m^2$. (in $Rs.$) [Use $\pi = \frac{22}{7}$] (Note that the base of the tent will not be covered with canvas.)

Mayank made a bird-bath for his garden in the shape of a cylinder with a hemispherical depression at one end (see figure). The height of the cylinder is $1.45\, m$ and its radius is $30\, cm$. Find the total surface area of the bird-bath in $m^2$. (Take $\pi = \frac{22}{7}$)

Difficult
View Solution

$A$ hemispherical depression is cut out from one face of a cubical wooden block such that the diameter $l$ of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid. $\left[\text{Use } \pi=\frac{22}{7}\right]$

$A$ spherical glass vessel has a cylindrical neck $8 \, cm$ long and $2 \, cm$ in diameter. The diameter of the spherical part is $8.5 \, cm$. By measuring the amount of water it holds,a child finds its volume to be $345 \, cm^{3}$. Check whether she is correct,taking the above as the inside measurements,and $\pi = 3.14$. (in $, cm^{3}$)

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial
For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free
For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo