gas is pumped into a spherical balloon at the rate of 1 cubic feet per minute. How fast is the diameter of the balloon increasing when the balloo0n contains 36 cubic feet of gas?

V = (4/3)πr^3

when V = 36 ...
36 = (4/3)πr^3
r^3 = 27/π
r = 2.04835

dV/dt = 4πr^2 dr/dt

1 = 4π(2.04835^2) dr/dt
dr/dt = .018966 ft^3/min

since diameter = 2r
d(diameter)/dt = .0379 ft^3 /min

first find dr/dt. The diameter changes twice as fast as the radius.

v = (4/3) pi r^3
dv/dt = 4 pi r^2 dr/dt
so
1 = 4 pi (r^2) dr/dt

but r^3 = (3/4)(36)/pi
so r = 2.05 ft
so
1 = 4 pi (4.2) dr/dt
so
dr/dt = .019 ft/min
D = 2 r
dD/dt = 2 dr/dt = .038 ft/min

another way
rate of volume increase = surface area * dr/dt
1 = 4 pi r^2 * dr/dt
same old equation

A spherical balloon is inflated so that its volume is increasing at the rate of 2.2 ft^3/min. How rapidly is the diameter of the balloon increasing when the diameter is 1.2 feet?

To solve this problem, we can use the concept of related rates. We need to find the rate at which the diameter of the balloon is increasing when it contains 36 cubic feet of gas.

Let's denote the diameter of the balloon as 'd' and the rate at which the diameter is changing as 'ddt'. We are given that the gas is pumped into the balloon at a rate of 1 cubic foot per minute, which we can represent as dV/dt = 1 ft^3/min.

We know that the volume of a sphere is given by V = (4/3)πr^3, where 'V' is the volume and 'r' is the radius of the sphere. Since the diameter is twice the radius, we can express the volume in terms of the diameter as V = (1/6)πd^3.

Differentiating both sides of this equation with respect to time (t), we get:
dV/dt = (1/6) * 3πd^2 * d(d/dt)
1 = (1/2)πd^2 * d(d/dt)
(d/dt) = 2 / (πd^2)

Now, we need to find the value of (d/dt) when the balloon contains 36 cubic feet of gas. Since V = 36 ft^3, we can substitute this into the equation to solve for (d/dt):
1 = (1/2)πd^2 * (2 / d(d/dt))
1 = πd * (1 / (d(d/dt)))
1 = π/((d(d/dt)))
π = (d(d/dt))

Therefore, (d(d/dt)) = π.

When the balloon contains 36 cubic feet of gas, we can substitute V = 36 ft^3 into the equation for the volume:
36 = (1/6)πd^3
d^3 = (6/π)(36)
d^3 = 216/π
d = (216/π)^(1/3)
d ≈ 6.893 ft

Now, we can calculate the rate at which the diameter is increasing by substituting the values we found into the equation (d/dt) = 2 / (πd^2):
(d/dt) = 2 / (π(6.893)^2)
(d/dt) ≈ 2 / 149.762
(d/dt) ≈ 0.0133 ft/min

Therefore, the diameter of the balloon is increasing at a rate of approximately 0.0133 feet per minute when the balloon contains 36 cubic feet of gas.