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Mathematics · 2023 · 4 marks
CBSE 2023 · Region 5 · Set 1 · Q36
A golf ball is spherical with about 300-500 dimples that help increase its velocity while in play. Golf balls are traditionally white but available…
A golf ball is spherical with about $\displaystyle 300$-$\displaystyle 500$ dimples that help increase its velocity while in play. Golf balls are traditionally white but available in colours also. In the given figure, a golf ball has diameter $\displaystyle 4 \cdot 2$ cm and the surface has $\displaystyle 315$ dimples (hemi-spherical) of radius $\displaystyle 2$ mm..
Based on the above, answer the following questions :
(i)
Find the surface area of one such dimple.
(ii)
Find the volume of the material dug out to make one dimple.
(iii)
(a)
Find the total surface area exposed to the surroundings.
OR
(b)
Find the volume of the golf ball.
Marking-scheme solution
(i)
$\displaystyle \mathrm{SA}=2 \pi \mathrm{r}^{2}=2 \times \frac{22}{7} \times 4=\frac{176}{7} \mathrm{~mm}^{2}$ or $\displaystyle 25 \cdot 1 \mathrm{~mm}^{2}$
(ii)
Volume of material dug out to make one dimple $\displaystyle =\frac{2}{3} \times \frac{22}{7} \times 8$
\[=\frac{352}{21} \mathrm{~mm}^{3} \text { or } 16.76 \mathrm{~mm}^{3}
\]
(iii)
radius of ball $\displaystyle =21 \mathrm{~mm}$
Total surface area exposed to surroundings
\[\begin{array}{l}
=4 \pi(21)^{2}-315 \times \pi(2)^{2}+315 \times 2 \pi(2)^{2} \\
=4 \times \frac{22}{7} \times 21 \times 21+\frac{22}{7} \times 315 \times 4 \\
=9504 \mathrm{~mm}^{2}
\end{array}
\]
Volume of the golf ball $\displaystyle =\frac{4}{3} \pi(21)^{3}-315 \times \frac{2}{3} \pi(2)^{3}$
\[=33528 \mathrm{~mm}^{3}
\]
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