Question

A cone of height 24 cm and radius of base 6 cm is made up of modelling clay. A child reshapes it in the form of a sphere. The radius of sphere is
(a) 1 cm
(b) 2 cm
(c) 4 cm
(d) 6 cm

Answer 1

Hint: Here, we have to equate the volumes of cone and sphere as the same clay is reshaped. i.e.
\[Volume\text{ }of\text{ }sphere\text{ =}Volume\text{ }of\text{ }cone\]
Then, apply the formulas:
\[\begin{align}
  & Volume\text{ }of\text{ }cone=\dfrac{1}{3}\pi {{r}^{2}}h \\
 & Volume\text{ }of\text{ }sphere=\dfrac{4}{3}\pi {{r}^{3}} \\
\end{align}\]

Complete step-by-step answer:
Here we are given the height and radius of the cone. They are:
Height of the cone, $ h=24\text{ }cm $
Radius of the cone, $ r=6\text{ }cm $
 Here, it is also given that a child reshapes cones in the form of a sphere. Therefore we can write:
\[Volume\text{ }of\text{ }sphere\text{ =}Volume\text{ }of\text{ }cone\]
 Now, we have to find the radius of the sphere. Let R be the radius of the sphere. Then, we have the equation:
 \[\dfrac{4}{3}\pi {{R}^{3}}=\dfrac{1}{3}\pi {{r}^{2}}h\text{ }....\text{ (1)}\]
 Next, by substituting the values of $ h=24 $ and $ r=6 $ in equation (1) we obtain:
\[\dfrac{4}{3}\pi {{R}^{3}}=\dfrac{1}{3}\pi {{6}^{2}}\times 24\]
 Now, by cross multiplication we get:
\[\dfrac{4}{3}\pi \times \dfrac{3}{\pi }{{R}^{3}}={{6}^{2}}\times 24\]
 In the next step, we have to cancel 3 and $ \pi $ we will get:
\[4{{R}^{3}}={{6}^{2}}\times 24\]
 Next, again by cross multiplication we get:
\[{{R}^{3}}=\dfrac{{{6}^{2}}\times 24}{4}\]
Now, again by cancellation we obtain:
\[\begin{align}
  & {{R}^{3}}={{6}^{2}}\times 6 \\
 & {{R}^{3}}={{6}^{3}} \\
\end{align}\]
Now, by taking cube root on both the sides we get:
\[\begin{align}
  & \sqrt[3]{{{R}^{3}}}=\sqrt[3]{{{6}^{3}}} \\
 & R=6 \\
\end{align}\]
Hence, we can say that the radius of the sphere, \[R=6\text{ }cm\]
Therefore, the correct answer for this question is option (d).

Note: Here, while calculating the volume make sure to take all the lengths in the same unit. If the units are different you have to convert it into any one unit and then find the answer.