# The total surface area of a solid hemisphere is $462c{{m}^{2}}$. Find its radius.

Last updated date: 24th Mar 2023

•

Total views: 306.3k

•

Views today: 5.83k

Answer

Verified

306.3k+ views

Hint: Use the formula for the total surface area of a solid hemisphere, that is $3\pi {{r}^{2}}$. Equate this with the given total surface area and solve the equation to find the radius.

Complete step-by-step answer:

A solid sphere has two surfaces, a curved surface and a flat surface. The total surface area of the hemisphere is the sum of the surface areas of both these surfaces. The curved surface area is given as $2{{\pi }^{2}}$ and the surface area of the flat surface at the bottom is given by the same formula as the area of a circle as it is circular in shape; that is, $\pi {{r}^{2}}$.

Thus, the total surface area of the solid hemisphere will be $2\pi {{r}^{2}}+\pi {{r}^{2}}=3\pi {{r}^{2}}$.

Equating this surface area with the total surface area given in the question, we get

$\begin{align}

& 3\pi {{r}^{2}}=462c{{m}^{2}} \\

& \Rightarrow \pi {{r}^{2}}=\dfrac{462}{3}c{{m}^{2}} \\

& \Rightarrow \pi {{r}^{2}}=154c{{m}^{2}} \\

\end{align}$

Using the value of $\pi =\dfrac{22}{7}$ in this equation we get

$\begin{align}

& \dfrac{22}{7}{{r}^{2}}=154c{{m}^{2}} \\

& \Rightarrow {{r}^{2}}=154c{{m}^{2}}\times \dfrac{7}{22} \\

& \Rightarrow {{r}^{2}}=49c{{m}^{2}} \\

\end{align}$

Solving the equation by taking the positive square root on both sides, we get $r=7cm$. Thus the radius of the given solid hemisphere is 7 cm.

Note: Since the hemisphere is solid, therefore the base area also needs to be considered and not only the curved surface. It is common to make this mistake of considering only the curved surface in calculation of the total surface area, and should be kept in mind while solving such questions.

Complete step-by-step answer:

A solid sphere has two surfaces, a curved surface and a flat surface. The total surface area of the hemisphere is the sum of the surface areas of both these surfaces. The curved surface area is given as $2{{\pi }^{2}}$ and the surface area of the flat surface at the bottom is given by the same formula as the area of a circle as it is circular in shape; that is, $\pi {{r}^{2}}$.

Thus, the total surface area of the solid hemisphere will be $2\pi {{r}^{2}}+\pi {{r}^{2}}=3\pi {{r}^{2}}$.

Equating this surface area with the total surface area given in the question, we get

$\begin{align}

& 3\pi {{r}^{2}}=462c{{m}^{2}} \\

& \Rightarrow \pi {{r}^{2}}=\dfrac{462}{3}c{{m}^{2}} \\

& \Rightarrow \pi {{r}^{2}}=154c{{m}^{2}} \\

\end{align}$

Using the value of $\pi =\dfrac{22}{7}$ in this equation we get

$\begin{align}

& \dfrac{22}{7}{{r}^{2}}=154c{{m}^{2}} \\

& \Rightarrow {{r}^{2}}=154c{{m}^{2}}\times \dfrac{7}{22} \\

& \Rightarrow {{r}^{2}}=49c{{m}^{2}} \\

\end{align}$

Solving the equation by taking the positive square root on both sides, we get $r=7cm$. Thus the radius of the given solid hemisphere is 7 cm.

Note: Since the hemisphere is solid, therefore the base area also needs to be considered and not only the curved surface. It is common to make this mistake of considering only the curved surface in calculation of the total surface area, and should be kept in mind while solving such questions.

Recently Updated Pages

If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

The coordinates of the points A and B are a0 and a0 class 11 maths JEE_Main

Trending doubts

Write an application to the principal requesting five class 10 english CBSE

Tropic of Cancer passes through how many states? Name them.

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE

What is per capita income

Change the following sentences into negative and interrogative class 10 english CBSE

A Short Paragraph on our Country India