Question

A golf ball has a radius of 4.2 cm. Its surface has 200 hemispherical dimples, each with a radius of 2.1 mm. Determine the total surface area that is exposed to the surroundings.

Answer 1

To calculate the total surface area of a golf ball, we first need to calculate the surface area of the ball as a sphere.

Then we need to add the curved surface area of all the 200 hemispherical dimples and subtract the common area between the two, that is the base area of all the 200 dimples.

Step 1: Calculate the surface area of the entire sphere.

We know that the formula for the surface area of a sphere with radius r is given by $4\pi r^2$.
Here $r=4.2cm$
Surface area of the sphere = $4\pi r²$
= $4\times \left(\frac{22}{7}\right)\times 4.2\times 4.2$
= $221.76c{m}^2$

Step 2: Calculate the curved surface area of each hemispherical dimple.

Curved surface area of a hemisphere with radius r is given by $2\pi r^2$
Here $r=2.1mm.$
Curved surface area of hemisphere = $2\pi r^2=2\times \left(\frac{22}{7}\right)\times 2.1\times 2.1$
= $27.72m{m}^2$
But we have 200 dimples on the golf ball, therefore CSA of all dimples = $200\times 27.72=5544m{m}^2$
= $55.44c{m}^2$

Step 3: Calculate the Base Area of the hemispherical dimples.

Base area of each dimple = $\pi r^2$
= $\left(\frac{22}{7}\right)\times 2.1\times 2.1$
= $13.86m{m}^2$
Base area of all 200 dimples = $200\times 13.86=2772m{m}^2$
= $27.72c{m}^2$

Step 4: Add the surface area of the sphere and CSA of the hemisphere and subtract the base area of the hemisphere.

The total area of the golf ball
= $221.76+55.44-27.72$
= $249.48c{m}^2$