Question

I. If the radius of a sphere is doubled. What will happen to its surface area?
II. Find the volume of a sphere whose surface area is \[154c{m^2}\] .

Answer 1

Hint: Radius is a line segment extending from the center of a circle or sphere to the circumference or bounding surface. We can find the Surface area of the sphere by \[4\pi {r^2}\] . The volume V of a sphere is four-thirds times pi times the radius cubed. The volume here depends on the diameter of radius of the sphere. The surface area of sphere is the area or region of its outer surface. To calculate the volume of a sphere, whose radius is ‘r’ we have: \[\dfrac{4}{3}\pi {r^3}\]
Formula used:
Surface area of the sphere = \[4\pi {r^2}\]
Volume of the sphere = \[\dfrac{4}{3}\pi {r^3}\]
In which, r is the radius of the sphere.

Complete step by step solution:
Let us solve:
i) If the radius of a sphere is doubled. What will happen to its surface area.
 \[ \to \] As given the radius of a sphere is doubled, hence
Radius is denoted as r.
We know that the Surface area of the sphere = \[4\pi {r^2}\]
If radius is doubled, then the radius can be written as 2r.
Replacing 2r in the area of surface formula i.e., \[4\pi {r^2}\] we get:
= \[4 \times \dfrac{{22}}{7} \times {\left( {2r} \right)^2}\]
Where, we know that value of \[\pi = \dfrac{{22}}{7}\] .
= \[4 \times \dfrac{{22}}{7} \times 4{r^2}\]
Hence, we get:
= \[16\pi {r^2}\]
= \[4 \times \] original surface area.
Therefore, the surface area becomes 4 times.
Hence, the surface area of the sphere will increase 4 times if the radius of that sphere is doubled.
Now, let us solve:
ii) Find the volume of a sphere whose surface area is \[154c{m^2}\] .
 \[ \to \] We know that the Surface area of the sphere = \[4\pi {r^2}\] = \[154c{m^2}\]
 \[4 \times \dfrac{{22}}{7}{r^2}\] = \[154c{m^2}\]
 \[ \Rightarrow \] \[{r^2} = \dfrac{{154}}{4} \times \dfrac{7}{{22}}\]
 \[ \Rightarrow \] \[{r^2} = \dfrac{{49}}{4}\]
 \[ \Rightarrow \] \[r = \sqrt {\dfrac{{49}}{4}} \]
Therefore, the value of radius r is:
 \[r = \dfrac{7}{2}\]
 \[r = 3.5cm\]
We need to find the volume of the sphere; hence we know that the volume of the sphere is given as:
Volume of the sphere = \[\dfrac{4}{3}\pi {r^3}\]
Now, substitute the value of \[\pi \] and the obtained value of r we get:
= \[\dfrac{4}{3} \times \dfrac{{22}}{7} \times {\left( {3.5} \right)^3}\]
= \[\dfrac{{88}}{{21}} \times 42.875\]
= \[179.67\;c{m^3}\]
So, the correct answer is “ \[179.67\;c{m^3}\] ”.

Note: We know that radius is half the diameter(D), in which the formula is given as \[r = \dfrac{D}{2}\] , this is identical to the method used for calculating the radius of a circle or sphere from its diameter, here in the given question the radius is doubled and we know that area of sphere is given as \[4\pi {r^2}\] , hence the area is four times the original surface area. We must note that, the volume of a sphere is \[\dfrac{2}{3}\] of the volume of a cylinder with the same radius, and height equal to the diameter.