To find the rate at which the radius of the balloon is increasing, we can use the formula for the volume of a sphere and differentiate it with respect to time.
Let's start by writing down the formula for the volume of a sphere:
V = (4/3) * π * r^3
Where V represents the volume and r represents the radius.
Now, we need to differentiate both sides of the equation with respect to time (t):
dV/dt = d/dt[(4/3) * π * r^3]
The left side gives us the rate at which the volume is changing, which is given in the problem as 75 cm^3/s:
dV/dt = 75
On the right side, we need to apply the product rule for differentiation:
d/dt[(4/3) * π * r^3] = (4/3) * π * d/dt(r^3)
Using the power rule for differentiation, we can find d/dt(r^3):
d/dt(r^3) = 3r^2 * dr/dt
Now we can substitute this back into the equation:
75 = (4/3) * π * 3r^2 * dr/dt
Simplifying, we get:
25 = 4Ï€r^2 * dr/dt
Now, we can solve for dr/dt, the rate at which the radius is changing:
dr/dt = 25 / (4Ï€r^2)
To find the rate at which the radius is increasing when the radius is 5 cm, we can substitute r = 5 into the equation:
dr/dt = 25 / (4Ï€ * 5^2)
dr/dt = 25 / (4Ï€ * 25)
dr/dt = 1 / (4Ï€)
Therefore, when the radius is 5 cm, the rate at which the radius is increasing is 1/(4Ï€) cm/s.
Now, let's show that one minute after the pumping begins, the radius is increasing at the rate of 1/(12Ï€ * r^(1/3)) per second.
To find the rate of change when one minute has passed, we'll need to convert one minute into seconds. Since there are 60 seconds in a minute, one minute is equal to 60 seconds.
Now we can substitute t = 60 into the equation for dr/dt:
dr/dt = 25 / (4Ï€ * r^2)
dr/dt = 25 / (4Ï€ * r^2) * t, where t is in seconds
dr/dt = 25 / (4Ï€ * r^2) * 60
dr/dt = 1500 / (4Ï€ * r^2)
Next, we'll substitute r = (1/12Ï€) into the equation:
dr/dt = 1500 / (4Ï€ * (1/12Ï€)^2)
dr/dt = 1500 / (4Ï€ * 1/(144Ï€^2))
dr/dt = 1500 / (4/144Ï€^3)
dr/dt = 1500 / (1/36Ï€^3)
dr/dt = 36Ï€^3 * 1500
dr/dt = 54000Ï€^3
So, one minute after the pumping begins, the rate at which the radius is increasing is 54000Ï€^3 cm/s, which can also be written as 1/(12Ï€ * r^(1/3)) per second.