Final Answer:
The radius is increasing at a rate of 0.1 ft/s after 3 minutes.
Step-by-step explanation:
To find the rate at which the radius is increasing, we can use the formula for the volume of a sphere, V = (4/3)πr^3, where V is the volume and r is the radius. We are given that helium is pumped into the balloon at a rate of 3 cubic feet per second. After 3 minutes, which is 180 seconds, the amount of helium pumped into the balloon would be 3 * 180 = 540 cubic feet. Using the formula for the volume of a sphere, we can set up an equation to solve for the rate at which the radius is increasing.
We have V = (4/3)πr^3 and dV/dt = 3 ft^3/s. To find dr/dt, we differentiate V with respect to t and solve for dr/dt. Differentiating V with respect to t gives us dV/dt = 4πr^2 * dr/dt. Plugging in the given values, we get 3 = 4πr^2 * dr/dt. Solving for dr/dt gives us dr/dt = 3 / (4πr^2). Substituting r = (3V / (4π))^(1/3), we can find the value of dr/dt after substituting V = 540 cubic feet.
After substituting V = 540 cubic feet into the equation, we get r ≈ 6.63 ft. Substituting this value into dr/dt = 3 / (4πr^2), we find that dr/dt ≈ 0.1 ft/s.