Final answer:
The rate of increase in volume of an expanding sphere depends on the factor by which the radius is increasing. To find the factor, we equate the change in volume to the given rate of increase and solve for the ratio of the new radius to the original radius. The rate of increase in volume is given by (12/r₁³ + 1)^(1/3).
Step-by-step explanation:
The rate of increase in volume of an expanding sphere is directly proportional to the original volume. The volume of a sphere is given by V = (4/3)πr³, where V is the volume and r is the radius. In this case, the rate of increase in volume is 12 cubic feet, so we need to find the factor by which the radius is increasing.
Let's assume the radius is increasing by a factor of x. The original volume is V₁ = (4/3)πr₁³, and the new volume is V₂ = (4/3)πr₂³. We know that V₂ - V₁ = 12 cubic feet. Substituting the formulas for volume and simplifying, we get (4/3)π(r₂³ - r₁³) = 12.
Since the rate of increase is the same for both radius and volume, we can cancel out the (4/3)π term and get r₂³ - r₁³ = 12. We want to find the factor by which the volume is increasing, so we divide both sides of the equation by r₁³. This gives us (r₂/r₁)³ - 1 = 12/r₁³. Solving for (r₂/r₁), we get (r₂/r₁) = (12/r₁³ + 1)^(1/3).
Therefore, the factor by which the volume is increasing is (12/r₁³ + 1)^(1/3). To find the rate of increase in volume, we substitute the given radius into the equation and calculate the value.