Question

TSA of hemisphere is $5940$ . Find its diameter.

Answer 1

Hint: We have given the total surface area of the hemisphere. We have to calculate the diameter of the hemisphere. Since diameter is two times the radius so firstly we calculate the value of radius of hemisphere. This can be done by using the formula of total surface area of hemisphere we equate the formula of total surface area of hemisphere with the given value of hemi-sphere and thus calculate radius. Once the radius is calculated we can find diameter by multiplying radius by two.

Complete step-by-step answer:
We have given the total surface area of the hemi-sphere. Total surface area of hemi-sphere is equal to $5940$.

We have to find the diameter of hemi-sphere
Let $r$ be the radius of the hemi-sphere.
So the total surface area of hemi-sphere is $3\pi {r^2}$
So $3\pi {r^2} = 5940$
Value of $\pi $ is $\dfrac{{22}}{7}$ so
$3 \times \dfrac{{22}}{7} \times {r^2} = 5940$
$ \Rightarrow {\text{ }}{{\text{r}}^2} = \dfrac{{5940 \times 7}}{{3 \times 22}}{\text{ }} \Rightarrow {\text{ }}{{\text{r}}^2} = \dfrac{{270}}{3} \times 7$
Dividing $270$ by $3$ we get
${{\text{r}}^2} = 90 \times 7$
${{\text{r}}^2} = 630$
Taking square root on both side
$\sqrt {{{\text{r}}^2}} = \sqrt {630} $
$r = \sqrt {630} $
Factors of $630 = 2 \times 3 \times 3 \times 5 \times 7$
So $r = \sqrt {2 \times 3 \times 3 \times 5 \times 7} $
$ = 3\sqrt {2 \times 3 \times 7} {\text{ }} \Rightarrow {\text{ }}3\sqrt {70} $
Radius of hemisphere $ = 3\sqrt {70} $
Diameter of hemisphere
$ = 2 \times radius$
$ = 2 \times 3\sqrt {70} $
$ = 6\sqrt {70} $

Note: When a sphere is divided into two equal parts each part is called hemi-sphere. There are two types of areas of the hemi-sphere. Curved surface area and total surface area. Surface area is the area of the circular part. Total surface area included both curved surface area and area of circular part. Diameter of a sphere is a straight line that passes through the centre of the sphere. It is also called the longest chord.