Question

A solid is composed of a cylinder with hemispherical ends. If the whole length of solid is 108 cm and the diameter of the hemispherical ends is 36 cm, find the cost of polishing its surface at the rate of 7 paise per sq. cm.

Answer 1

Hint: Determine the length of the cylinder and then the curved surface area of the cylinder, that is, \[2\pi rh\] and the outer surface area of the hemispherical ends, that is, \[2\pi {r^2}\] each and then multiply the total surface area with 10 to get the desired answer.

Complete step-by-step answer:

We need to find the cost of polishing the solid composed of a cylinder with hemispherical ends.
Hence, we need to find the total surface area of the solid. It is equal to the sum of the curved surface area of the cylinder and surface area of the two hemispheres.

The total length of the solid is the sum of the length of the cylinder and the diameter of the hemispheres.
Then, the length of the cylinder part alone is given by:
\[{h_C} = 108 - 36\]
\[{h_C} = 72cm\]
The radius of the solid is given by,
\[r = \dfrac{d}{2}\]
\[r = \dfrac{{36}}{2}\]
\[r = 18cm\]
We know that the curved surface area of a cylinder with radius r and length h is given as follows:
\[S = 2\pi rh.........(1)\]
The curved surface area of the cylinder with radius 18 cm and length 72 cm is given by equation (1) as follows:
\[{S_C} = 2 \times \dfrac{{22}}{7} \times 18 \times 72\]
\[{S_C} = \dfrac{{57024}}{7}c{m^2}..............(2)\]
We, now, find the surface area of the hemispherical ends.
The surface area of a hemisphere is given by:
\[S = 2\pi {r^2}...........(3)\]
The surface area of two hemispherical ends of radius 18 cm is given as follows:
\[{S_H} = 2 \times 2 \times \dfrac{{22}}{7} \times {(18)^2}\]
\[{S_H} = \dfrac{{28512}}{7}c{m^2}................(4)\]
The total surface area of the solid is then the sum of equation (3) and equation (4).
\[S = {S_C} + {S_H}\]
\[S = \dfrac{{57024}}{7} + \dfrac{{28512}}{7}\]
\[S = \dfrac{{85536}}{7}c{m^2}\]
The cost of polishing is 7 paise per sq. cm.
Hence, the total cost of painting is given as follows:
Total cost = \[\dfrac{{85536}}{7} \times 7\]
Total cost = 85536 paise
Total cost = Rs. 855.36
Hence, the total cost of painting the solid is Rs. 855.36.

Note: Note that the total length of the solid is 108 cm and not the length of the cylindrical part alone. Hence, calculate the length of the cylindrical part first before calculating its curved surface area.