Consider the following statements in respect of four spheres A, B, C and D having respective radii 6, 8, 10 and 12 cm :
1. The surface area of sphere C is equal to the sum of surface areas of spheres A and B.
2. The volume of sphere D is equal to the sum of volumes of spheres A, B and C.
Which of the above statements is/are correct?
Correct Answer: Description for Correct answer:
Let the four sphere A, B, C and D have radii, 6, 8, 10 and 12 cm respectively
Statement I
Surface area of sphere (C)
= \( \Large 4 \pi r_{c}^{2} \)
=\( \Large 4 \pi (10)^{2} \)
\( \Large 400 \pi \) ... (I)
Now sum of surface area of sphere A and B
= \( \Large 4 \pi (r_{a}^{2} + r_{b}^{2}) \)
= \( \Large 4 \pi (36 + 64) \)
= \( \Large 400 \pi \) ...(ii)
From (i) and (ii), it can conclude that statement-I is correct
Statement II:
Volume of sphere(D) = \( \Large \frac{4}{3} \pi (12)^{3} \)
= \( \Large \frac{4}{3} \pi (1728) \)... (i)
Now sum of volumes of spheres A, B and C
=\( \Large \frac{4}{3} \pi \left( 6^{3} + 8^{3} + 10^{3}\right) \)
=\( \Large \frac{4}{3} \pi \left( 216 + 512 + 1000 \right) \)
=\( \Large \frac{4}{3} \pi \left( 1728 \right) \) .. (ii)
From (i) and (ii) it can be concluded that statement-ii is also correct
Option (c) is correct
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