A bucket is open at the top, and made up of a metal sheet is in the form of a frustum of a cone. The depth of the bucket is 24 cm and the diameters of its upper and lower circular ends are 30 cm and 10 cm respectively. Find the cost of metal sheet used in it at the rate of Rs. 10 per $100\ cm^{2}$. [use $\pi =3.14$].

Given: A frustum shaped bucket with the depth of 24 cm and diameter of its upper and lower circular ends 30 cm and 10cm respectively. rate of metal sheet=Rs. $\frac{10}{1000}$ cm$^{2}$. 

To do: To find the cost of metal sheet used in making the frustum.

Solution:

Diameter of upper end of bucket, $d_{1} =30$ cm

Radius $( r_{1}) \ of\ upper\ end\ of\ bucket\ =\frac{d_{1}}{2}$

$=\frac{30}{2}$

$=15$ cm

Diameter of lower end of bucket $d_{2}=10$ cm

Radius $( r_{2}) \ of\ lower\ end\ of\ bucket=\frac{d_{2}}{2}$

$=\frac{10}{2}$

$=5$ cm

Slant height $( l)$  of frustum $=\sqrt{( r_{2} -r_{1})^{2} +h^{2}}$

$=\sqrt{( 15-5)^{2} +24^{2}}$

$=\sqrt{100+576}$

$=\sqrt{676}$

$=26$ cm

Area of metal sheet used to make the bucket$=$Area of the frustum$+$Area of the its base

$=\pi ( r_{1} +r_{2}) l+\pi r^{2}_{2}$

$=\pi \left[( 15+5) 26+5^{2}\right]$

$=\pi [( 20\times 26) +25]$

$=545\pi \ cm^{2}$

Cost of $100\ cm^{2}$ metal sheet$=Rs.10$

Cost of $545\pi \ cm^{2}$ metal sheet $=\frac{545\ \times 3.14\times 10}{100}$

$=Rs.\ 171.13$

Therefore, Cost of metal sheet used to make the bucket is Rs.171.13.
 

Updated on: 10-Oct-2022

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