4. Five mark exercise problems I

1. The outer and the inner surface areas of a spherical copper shell are 576\(\pi\) \(cm^2\) and 324\(\pi\) \(cm^2\) respectively. Find the volume of the material required to make the shell.

 

Answer:

 

The volume of the material required \(=\) \(cm^3\)

 

 

2. A solid sphere and a solid hemisphere have equal total surface area. Prove that the ratio of their volume is \(3 \sqrt{3}:4\).

 

Proof:

 

Let \(r_1\) be the radius of a sphere and \(r_2\) be the radius of a hemisphere.

 

 

 

Ratio of their volumes \(=\) $\frac{\left(\frac{i}{i}\mathrm{\pi }{r_1}^i\right)}{\left(\frac{i}{i}\mathrm{\pi }{r_2}^i\right)}$

 

\(=\) $\frac{i{r_1}^i}{{r_2}^i}$

 

\(=\) $\frac{i\times {\left(\sqrt{i}\right)}^i}{i^i}$

 

\(=\) $\frac{i\sqrt{i}}{i}$

 

Hence, proved.

 

 

3. A hemi-spherical hollow bowl has material of volume $\frac{436\mathrm{\pi }}{3}$ cubic cm. Its external diameter is  \(14 \ cm\). Find its thickness.