a spherical snowball is melting in such a way that its radius is decreasing at rate of 0.3 cm/min. at what…

a spherical snowball is melting in such a way that its radius is decreasing at rate of 0.3 cm/min. at what rate is the volume of the snowball decreasing when the radius is 12 cm. (note the answer is a positive number).\n\nhint: the volume of a sphere of radius r is ( v=\frac{4}{3}pi r^{3} )

a spherical snowball is melting in such a way that its radius is decreasing at rate of 0.3 cm/min. at what rate is the volume of the snowball decreasing when the radius is 12 cm. (note the answer is a positive number).\n\nhint: the volume of a sphere of radius r is ( v=\frac{4}{3}pi r^{3} )

Answer

Explanation:

Step1: Differentiate the volume formula

The volume of a sphere is (V=\frac{4}{3}\pi r^{3}). Differentiate (V) with respect to time (t) using the chain - rule. (\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}). First, find (\frac{dV}{dr}): (\frac{dV}{dr}=\frac{4}{3}\pi\cdot3r^{2}=4\pi r^{2}).

Step2: Substitute the given values

We know that (\frac{dr}{dt}=- 0.3) (negative because the radius is decreasing) and (r = 12). Substitute into (\frac{dV}{dt}=4\pi r^{2}\cdot\frac{dr}{dt}). (\frac{dV}{dt}=4\pi(12)^{2}\cdot(-0.3)). Calculate (4\pi\times144\times(- 0.3)=-172.8\pi). But the problem asks for a positive number (since it asks for the rate at which the volume is decreasing, and we can take the absolute value).

Answer:

(172.8\pi\approx542.87)