# If the total surface area of a solid hemisphere is \[462\text{ c}{{\text{m}}^{2}}\], find its volume. Take $\pi =\dfrac{22}{7}$.

(A) $716$

(B) $718.67$

(C) $720.87$

(D) $840$

Answer

Verified

327.3k+ views

Hint: Assume that the radius of the hemisphere is $r$. Calculate the value of $r$ from the given information about the total surface area (T.S.A.). T.S.A. of a solid hemisphere is $=3\pi {{r}^{2}}$. From here the value of radius $r$ can be calculated. Now, use volume $=\dfrac{2}{3}\pi {{r}^{3}}$ for the calculation of volume.

Complete step-by-step answer:

A solid hemisphere is obtained when we cut a solid sphere into two equal halves. The volume of the hemisphere is half that of the volume of the sphere.

Now, we have been given that the total surface area of the hemisphere is \[462\text{ c}{{\text{m}}^{2}}\].

We know that the lateral surface area of the hemisphere is $2\pi {{r}^{2}}$. In the solid hemisphere its circular base is present whose area is $\pi {{r}^{2}}$. Hence the total surface area of solid hemisphere is $2\pi {{r}^{2}}+\pi {{r}^{2}}=3\pi {{r}^{2}}$.

Since, total surface area$=462\text{ c}{{\text{m}}^{2}}$.

\[\begin{align}

& \therefore 3\pi {{r}^{2}}=462 \\

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

& \text{ }=\dfrac{462}{3\times \dfrac{22}{7}} \\

& \text{ }=\dfrac{462\times 7}{3\times 22} \\

& \text{ }=49 \\

& \therefore r=\sqrt{49}=7. \\

\end{align}\]

Now, to calculate the volume $V$, we use the formula:

\[\begin{align}

& V=\dfrac{2}{3}\pi {{r}^{3}}. \\

& \therefore V=\dfrac{2}{3}\times \dfrac{22}{7}\times {{7}^{3}} \\

& \text{ }=\dfrac{2}{3}\times \dfrac{22}{7}\times 7\times 7\times 7 \\

& \text{ =}718.67\text{ c}{{\text{m}}^{3}}. \\

\end{align}\]

Hence, option (B) is correct.

Note: We have used the given information to find the unknown variable which in this question is the radius $r$ of the solid hemisphere. Once the value of the unknown variable is determined, it is applied in the formula of the volume to get the answer. Also, note that cutting the sphere into two halves does not make the total surface area of the hemisphere half that of the sphere like in case of volume.

Complete step-by-step answer:

A solid hemisphere is obtained when we cut a solid sphere into two equal halves. The volume of the hemisphere is half that of the volume of the sphere.

Now, we have been given that the total surface area of the hemisphere is \[462\text{ c}{{\text{m}}^{2}}\].

We know that the lateral surface area of the hemisphere is $2\pi {{r}^{2}}$. In the solid hemisphere its circular base is present whose area is $\pi {{r}^{2}}$. Hence the total surface area of solid hemisphere is $2\pi {{r}^{2}}+\pi {{r}^{2}}=3\pi {{r}^{2}}$.

Since, total surface area$=462\text{ c}{{\text{m}}^{2}}$.

\[\begin{align}

& \therefore 3\pi {{r}^{2}}=462 \\

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

& \text{ }=\dfrac{462}{3\times \dfrac{22}{7}} \\

& \text{ }=\dfrac{462\times 7}{3\times 22} \\

& \text{ }=49 \\

& \therefore r=\sqrt{49}=7. \\

\end{align}\]

Now, to calculate the volume $V$, we use the formula:

\[\begin{align}

& V=\dfrac{2}{3}\pi {{r}^{3}}. \\

& \therefore V=\dfrac{2}{3}\times \dfrac{22}{7}\times {{7}^{3}} \\

& \text{ }=\dfrac{2}{3}\times \dfrac{22}{7}\times 7\times 7\times 7 \\

& \text{ =}718.67\text{ c}{{\text{m}}^{3}}. \\

\end{align}\]

Hence, option (B) is correct.

Note: We have used the given information to find the unknown variable which in this question is the radius $r$ of the solid hemisphere. Once the value of the unknown variable is determined, it is applied in the formula of the volume to get the answer. Also, note that cutting the sphere into two halves does not make the total surface area of the hemisphere half that of the sphere like in case of volume.

Last updated date: 06th Jun 2023

â€¢

Total views: 327.3k

â€¢

Views today: 2.83k

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