Question

1. If the curved surface area of the solid sphere is $$98.56c{m}^2$$ then find the radius of the sphere.
2. Surface area of the solid hemisphere is 2772sq.cm, then find its total surface area.

Answer 1

Hint: In this problem, we have to find the radius of the sphere whose total surface area given is $$98.56c{m}^2$$. For the first problem. We know that the formula for the curved surface area of a sphere is $$4\pi r^2$$, where r is the radius. We are already given the value of the curved surface area, which we have to equate to the formula and we can simplify it to get the value of the radius. For the second problem, we are given the surface area of the solid hemisphere from which we can find the radius value and substitute in the formula of total surface area to get the answer.

Complete step by step solution:
1.Here we have to find the radius of the sphere whose total surface area given is $$98.56c{m}^2$$.
We know that the formula for the curved surface area of a sphere is $$4\pi r^2$$, where r is the radius.
We are already given the value of the curved surface area, which we have to equate to the formula, we get
$$\Rightarrow 4\pi r^2=98.56$$
We can write the $$\pi ={\displaystyle \frac{22}{7}}$$ and simplify the above step, we get
 $$\begin{array}{rl}& \Rightarrow 4\times {\displaystyle \frac{22}{7}}{r}^2=98.56\\ & \Rightarrow r^2={\displaystyle \frac{98.56\times 7}{4\times 22}}={\displaystyle \frac{8.96\times 7}{8}}\\ & \Rightarrow r^2=1.12\times 7=7.84\end{array}$$
We can now take square root on both sides, we get
$$\Rightarrow r=\sqrt{7.84}=2.8cm$$

Therefore, the radius is 2.8cm.
2. Here we have to find the total surface area of a solid hemisphere whose curved surface area is 2772sq.cm.
We know that Curved surface area is $$2\pi r^2$$,
$$\begin{array}{rl}& \Rightarrow 2\pi r^2=2772\\ & \Rightarrow r^2=2772\times {\displaystyle \frac{7}{22\times 2}}\\ & \Rightarrow r^2=441\end{array}$$
 We can now take square root on both sides we get
$$\Rightarrow r=\sqrt{441}=21$$
We know that the total surface area of the hemisphere formula is $$3\pi r^2$$.
We can now substitute the value of r in the above formula, we get’
$$\begin{array}{rl}& \Rightarrow 3\times {\displaystyle \frac{22}{7}}\times {\left(21\right)}^2=3\times {\displaystyle \frac{22}{7}}\times 441=4158\end{array}$$

Therefore, the total surface area of the hemisphere is 4158sq.cm.

Note: We should always remember the formulas such as the total surface area of hemisphere formula is $$3\pi r^2$$, Curved surface area is $$2\pi r^2$$ and the curved surface area of a sphere is $$4\pi r^2$$, where r is the radius. We should know some of the square terms to be substituted directly.