Question : A solid cone of height 42 cm with a diameter of its base of 42 cm is cut out from a wooden solid sphere of radius 24 cm. Find the percentage of wood wasted correct to two places of decimal.
Option 1: 75.56%
Option 2: 56.65%
Option 3: 66.50%
Option 4: 67.50%
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Correct Answer: 66.50%
Solution :
Given, the height of the cone, $h_c$ = 42 cm
Radius of cone, $r_c$ = $\frac{42}{2}$ cm = 21 cm
Radius of sphere, $r_s$ = 24 cm
Volume of sphere = $\frac{4}{3} \pi r_s^3$
= $\frac{4}{3} \pi \times{24}^3$
= $18432\pi$ cm
3
Volume of cone = $ \frac{\pi r_c^2 h_c}{3}$
= $ \frac{\pi\times 21^2\times 42}{3}$
= $6174\pi$ cm
3
Wood wasted = Volume of Sphere – Volume of cone
= $18432\pi-6174\pi$
= $12258\pi$ cm
3
Percentage of wood wasted = $
\frac{\text{Volume of Sphere – Volume of cone}}{\text{Volume of Sphere}}\times 100$
= $\frac{12258\pi}{18432\pi}\times 100$
= 66.50%
Hence, the correct answer is 66.50%.
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