A balloon (in a shape of a sphere) is inflated with helium at a constant rate of 125 cm^3/s. Calculate the rate of increase of the diameter when the volume is 2000 cm^3.
V = (4/3)π r^3
dV/dt = 4π r^2 dr/dt
when V = 2000
(4/3)πr^3 = 2000
r^3 = 1500/π
r = (1500/π)^(1/3)
= 7.8159... I stored in my calculator's memory
so in
dV/dt = 4π r^2 dr/dt
125 = 4π (7.8159...)^2 dr/dt
dr/dt = .1628618
rate of change of diameter = .3225536 cm/s
To calculate the rate of increase of the diameter, we need to relate it to the rate of change of the volume. Given that the balloon is in the shape of a sphere, we can use the formula for the volume of a sphere to establish this relationship:
Volume of a sphere = (4/3) * π * r^3
Where r is the radius of the sphere.
First, we need to find the radius when the volume is 2000 cm^3. We can rearrange the volume formula to solve for the radius:
r = ((3 * volume) / (4 * π))^(1/3)
Substituting the value for volume, we have:
r = ((3 * 2000) / (4 * π))^(1/3)
r ≈ 7.617 cm
Now, we can differentiate the volume formula with respect to time to determine the rate of change of the volume:
dV/dt = (4/3) * π * (3 * r^2) * dr/dt
Where dV/dt is the rate of change of volume, and dr/dt is the rate of change of the radius.
Given that dV/dt = 125 cm^3/s, we can substitute the known values into the equation and solve for dr/dt:
125 = (4/3) * π * (3 * (7.617)^2) * dr/dt
dr/dt ≈ 0.0868 cm/s
Therefore, the rate of increase of the diameter is approximately 0.0868 cm/s.