The rate of change (in cm2/s) of the total surface area of a hemisphere with respect to radius r at \(r=\sqrt[3]{1.331}\) cm is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(6.6\pi\)
The total surface area of a hemisphere is the curved surface plus the flat circular base:
\(S = 2\pi r^2 + \pi r^2 = 3\pi r^2\)
Its rate of change with respect to r is
\( \displaystyle \frac{dS}{dr} = 6\pi r \)
Since \(1.1^3 = 1.331\):
\(r = \sqrt[3]{1.331} = 1.1\)
\( \displaystyle \frac{dS}{dr} = 6\pi \times 1.1 = 6.6\pi \)
Tip: The total surface area includes the base. Using only the curved surface \(2\pi r^2\) gives \(4\pi r = 4.4\pi\), Option (4).