Related Rates problem

Question

The radius of a spherical watermelon is growing at a constant rate of 2 centimeters per week. The thickness of the rind is always one tenth of the radius. The volume of the rind is growing at the rate ____  cubic centimeters per week at the end of the fifth week. Assume that the radius is initially zero.

Answer 1

So, for this problem, all we have is the rate at which the radius of the watermelon is growing (which includes the rind) as well as an expression for the radius of just the rind.

The formula for the volume of a sphere is V=(4/3)πr³. If we want to get a formula for the volume of just the rind of the watermelon, which is just an outer shell of the sphere, we would want to take the volume of the entire watermelon and subtract from it the volume of the watermelon which does not include the rind. We know that the thickness of the rind is .1 of the radius of the entire watermelon, which means the radius of the inner watermelon without the rind is .9r. We must include r here because we do not know what r is yet.

So, from that, the volume for the inner watermelon without the rind is V=(4/3)π(.9r)^3.

Then, from all this, we can get that the formula for the volume of just the rind is

V=(4/3)πr³-(4/3)π(.9r)^3.

We can condense this more to V=(4/3)πr³(1-(.9)^3).

This is the volume of the rind. Now, we implicitly differentiate

dV/dt = dV/dr * dr/dt

Using the chain rule, we get

dV/dt = 4πr²(1-(.9)³) * dr/dt

We have been given dr/dt in the problem, which is 2cm/week. Further, we know that we are searching for this at the end of the 5th week. This gives us both our r value and our dr/dt value for the equation. dr/dt=2, and r at the 5th week, given that it grows constantly at 2cm/week, is 2cm * 5 = 10 cm

Plugging this all in gives

dV/dt = 4π(10)²(1-(.9)³) * 2 = 681.097 cm. Using significant figures, the smallest figure we're able to measure is to the ones place. So, this rounds to 681 cm³/week, which is the rate at which the volume of the rind of the watermelon is growing at the end of the 5th week.

I hope this is clear for you and answers everything!