Question

What will be the inner and outer radius of a spherical shell of inner surface area $452.39c{{m}^{2}}$ and outer surface area $804.25c{{m}^{2}}$ ? (surface areas are accurate up to 2 decimal places).
(a) 6 cm, 7 cm
(b) 7 cm, 8 cm
(c) 6 cm, 8 cm
(d) 7 cm, 9 cm

Answer 1

Hint: By basic knowledge of geometry you can say the surface area of a sphere is given by $4\pi {{r}^{2}}$ , where r is the radius of the sphere. Here, we have a shell which is a combination of 2 shells with filled material in between. So, the areas of 2 shells are given and we are asked to find the radius of both shells. By equating the area given to the formula above the only remaining variable will be radius. So, by solving the equation we can find the radius of both the inner and outer sphere in a similar manner.

Complete step by step solution:
A spherical shell is a three-dimensional figure with hollow space in the middle and bounded by 2 spheres with different radii. The set of points which are in 3-D space and equal distance from a point will form the sphere.
Here, there are 2 radii of inner and outer spheres.
By basic knowledge of geometry, we can say area of sphere of radius r I given by the formula:
$A=4\pi {{r}^{2}}$
Given in question the inner sphere surface area $=452.39c{{m}^{2}}$
Given in question the outer sphere surface area $=804.25c{{m}^{2}}$
By assuming inner sphere radius to be denoted by r.
By assuming outer sphere radius to be denoted by R.
Now by the formula given above and inner surface area, we get:
$4\pi {{r}^{2}}=452.39c{{m}^{2}}$
By dividing with $4\pi $ on both sides of equation, we get:
${{r}^{2}}=\dfrac{452.39}{4\pi }$
On substituting value of $\pi =3.14$ and further solving, we get
${{r}^{2}}=\dfrac{452.39}{4\times 3.14}$
${{r}^{2}}=\dfrac{452.39}{12.56}$
${{r}^{2}}=36.01$
So, we can consider this as 36.
By applying square root on both sides of equation, we get
$r=6\text{cm}$
By applying square root on both sides of equation, we get
$r\approx 6\text{cm}$
Similarly, for outer spherical shell, we apply condition, we get:
$\text{4}\pi {{\text{R}}^{2}}=804.25$
By dividing with $4\pi $ and applying square root on both sides, we get:
R $=7.99cm\text{ }\approx \text{8cm}$
By above equations we can say values of inner and outer radii to be 6 cm, 8 cm.
Therefore, option (c) is correct.

Note: Here after the square root you should not consider the negative root as radius is always positive, as we use $\pi $ value nearly 3.14. So, we don’t get exact answers.