Answer
Verified
478.5k+ 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
Change the following sentences into negative and interrogative class 10 english CBSE
A Paragraph on Pollution in about 100-150 Words
One cusec is equal to how many liters class 8 maths CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What were the social economic and political conditions class 10 social science CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE
Trending doubts
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
How do you graph the function fx 4x class 9 maths CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE
What is the past participle of wear Is it worn or class 10 english CBSE
Why did the British treat the Muslims harshly immediately class 10 social science CBSE
A Paragraph on Pollution in about 100-150 Words