Question

How do you calculate the radius of a hemispherical solid whose total surface area is \[48\pi {\text{ }}c{m^2}\]?

Answer 1

Hint: We have to find the radius of a hemispherical solid whose total surface area is \[48\pi {\text{ }}c{m^2}\]. For this we will first assume the radius of the hemispherical solid to be \[r\]. Then using the formula of total surface area of a hemisphere i.e., \[3\pi {\left( {radius} \right)^2}\] we will calculate the total surface area in terms of \[r\] and then we will equate this total surface area to given total surface area i.e., \[48\pi {\text{ }}c{m^2}\] to find \[r\].

Complete step by step answer:

A hemisphere is the half piece of a sphere. If we cut the sphere into two equal parts, each part will have a curved surface as well as a flat surface.
The total surface area of a hemisphere is the sum of the curved surface area and flat surface area.
Therefore, \[{\text{total surface area of a hemisphere = flat surface area + curved surface area}}\]
Then, \[{\text{Total surface area of a hemisphere}} = \pi {\left( {radius} \right)^2} + 2\pi {\left( {radius} \right)^2}\]
\[\therefore {\text{Total surface area of a hemisphere}} = 3\pi {\left( {radius} \right)^2}\]
Let the radius of hemispherical solid be \[r\].
Given, total surface area of the hemispherical solid \[ = 48\pi {\text{ }}c{m^2}\]
Therefore, we can write
\[ \Rightarrow 3\pi {r^2} = 48\pi {\text{ }}\]
Dividing both the sides by \[3\pi \],
\[ \Rightarrow {r^2} = 16\]
On simplifying we get
\[ \Rightarrow r = 4{\text{ }}cm\]
Therefore, the radius of a hemispherical solid whose total surface area is \[48\pi {\text{ }}c{m^2}\] is \[4{\text{ }}cm\].

Note:
A hemisphere is half of a sphere, the total surface area of the hemisphere consists of two parts, which are the curved and flat surface areas. The curved surface area of a hemisphere is half of the surface area of a sphere i.e., \[2\pi {r^2}\] whereas in the total surface area of a hemisphere we additionally need to determine the surface area of the circular surface of the flat base of the given hemisphere, which is equal to \[\pi {r^2}\], which makes the total surface area to be \[3\pi {r^2}\].