Question

Your neighbor is growing a slightly different watermelon. It also has a rind whose thickness is one-tenth of the radius of that watermelon. However, the rind of your neighbor's watermelons grows at a constant rate of 20 cubic centimeters a week. The radius of your neighbor's watermelon after 5 weeks is ________ centimeters and at that time it is growing at _______ centimeters per week.

Answer 1

Let r be the radius of the watermelon in centimeters, and let t be the time in weeks. The volume of the watermelon can be expressed as:

V = (4/3) * π * (r - 0.1r)^3

Simplifying this expression, we get:

V = (4/3) * π * (0.9r)^3

V = (4/3) * π * 0.729r^3

The rate of change of the volume of the watermelon with respect to time can be expressed as:

dV/dt = (4/3) * π * 0.729 * 3r^2 * dr/dt

Simplifying this expression and substituting the given rate of change of the volume (20 cubic centimeters a week), we get:

20 = (4/3) * π * 0.729 * 3r^2 * dr/dt

Simplifying this expression, we get:

dr/dt = 20 / (3.1416 * 0.729 * 3 * r^2)

dr/dt = 0.765 / r^2

Now we can substitute t = 5 and solve for r and dr/dt:

r = sqrt(20/3.1416) = 2.52 centimeters (rounded to two decimal places)

dr/dt = 0.765 / (2.52)^2 = 0.122 centimeters per week (rounded to three decimal places)

Therefore, after 5 weeks, the radius of the watermelon is approximately 2.52 centimeters, and it is growing at a rate of approximately 0.122 centimeters per week