Answer :
Final answer:
The problem is an application of related rates in calculus. The formula for the rate of change of the surface area of the sphere is 4πr cm²/sec, not matching any of the given options.
Explanation:
To resolve this problem, we need to apply the concept of related rates in calculus. You're required to calculate how fast the surface area of a sphere is changing at the specific instant when the radius is increasing at a rate of 2 cm/second.
The formula for the surface area of the sphere is given as S=πr². However, we need to find ds/dt, the rate of increase of the Surface area. So, let's differentiate S with respect to time 't'. After differentiating, we obtain the equation dS/dt = 2πr*dr/dt, where dr/dt is the rate of change of radius which is given as 2 cm/sec. We need to plug the value of dr/dt into the differentiated equation to find the rate of change of the sphere's surface area.
After calculations, we get dS/dt = 4πr cm²/sec. From the list of options, none of them matches the correct answer, so there appears to be an error in the provided options as they fall short of the correct answer.
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