Final answer:
To find the rate at which the radius is increasing when the sphere contains 50 cubic inches of clown breath, we can set up an equation using the volume formula for a sphere. Rearranging the equation and substituting the given volume, we can solve for the radius and calculate the rate of increase.
Step-by-step explanation:
We can use the formula for the volume of a sphere to solve this problem. The formula is:
V = (4/3)πr^3
where V is the volume and r is the radius of the sphere.
Given that the balloon is blowing up at a rate of 100 cubic inches per minute, we can set up an equation to find the rate at which the radius is increasing. Let's call it dr/dt:
100 = (4/3)πr^2(dr/dt)
To find dr/dt, we need to solve for it. Rearranging the equation, we get:
dr/dt = 100 / [(4/3)πr^2]
Substituting the given volume of 50 cubic inches, we can solve for the radius using the volume formula:
50 = (4/3)πr^3
Simplifying, we get:
r = (3/(4π))^(1/3) * 50^(1/3)
Finally, substituting the value of r back into the equation for dr/dt, we can find the rate at which the radius is increasing when the sphere contains 50 cubic inches of clown breath.