Question

If the sphere has the same curved surface area as total surface area of cone of vertical height 40cm and radius 30cm, then the radius of the sphere is:
(a) $10\sqrt{6}cm$
(b) $10\sqrt{3}cm$
(c) $10\sqrt{2}cm$
(d) 12cm

Answer 1

Hint: We know that the total surface area of the cone is $\pi rl+\pi {{r}^{2}}$ , where l is equal to $\sqrt{{{r}^{2}}+{{h}^{2}}}$ . So, substitute the values in the formulas and equate it with the curved surface area of the sphere which is equal to $4\pi $ times the square of its radius. Solve the equation you get to get the radius of the sphere.

Complete step-by-step answer:
Let us start the solution to the above question by drawing the representative diagram of the figures given in the question.

We know that the total surface area of the cone is $\pi rl+\pi {{r}^{2}}$ , where l is equal to $\sqrt{{{r}^{2}}+{{h}^{2}}}$ . It is given in the question that the height of the cone is 40cm and the radius of the cone is 30cm.
$\text{TSA of cone}=\pi rl+\pi {{r}^{2}}=\pi r\sqrt{{{r}^{2}}+{{h}^{2}}}+\pi {{r}^{2}}$
$\Rightarrow \text{TSA of cone}=\pi \times 30\sqrt{{{30}^{2}}+{{40}^{2}}}+\pi \times {{30}^{2}}$
Now, we know that the square of 30 and 40 is 900 and 1600, respectively.
$\text{TSA of cone}=\pi \times 30\left( \sqrt{900+1600}+30 \right)$
$\Rightarrow \text{TSA of cone}=\pi \times 30\left( \sqrt{2500}+30 \right)$
We also know that the square root of 2500 is 50. If we use this in our equation, we get
$\text{TSA of cone}=\pi \times 30\left( 50+30 \right)=\pi \times 30\times 80=2400\pi \text{ c}{{\text{m}}^{2}}..............(i)$
Now, let us move to the curved surface area of the sphere. We know that the curved surface area of the sphere is $4\pi $ times the square of its radius. Also, it is given to be equal to the total surface area of the cone that we found in equation (i).
$4\pi {{\left( radius \right)}^{2}}=2400\pi $
\[\Rightarrow {{\left( radius \right)}^{2}}=\dfrac{2400\pi }{4\pi }\]
\[\Rightarrow {{\left( radius \right)}^{2}}=600\]
Now, if we take the square root of both the sides of the equation, we get
\[radius=\sqrt{600}=10\sqrt{6}\text{ cm}\]
Hence, the answer to the above question is option (a).

Note: The question is purely based on formulas. If you know the formulas of the curved surface area of the 3 dimensional figure mentioned in the question, then you just have to put the values and get the answer, so try to learn all the basic formulas related to 3 dimensional figures. Also, make sure that you don’t commit a calculation mistake. Also, remember that the curved surface area and the total surface area of a sphere is the same.