A balloon is being filled with helium at the rate of 4 ft^3/min. The rate, in feet per minute, at which the radius is increasing when the radius is 2 feet is (V= 4/3πr^3)
To find the rate at which the radius is increasing, we can use the related rates formula:
dV/dt = (4/3)π(3r^2)(dr/dt)
Where:
dV/dt is the rate at which the volume (V) is changing
r is the radius
dr/dt is the rate at which the radius is changing
Given:
dV/dt = 4 ft^3/min
r = 2 ft
We can plug in these values and solve for dr/dt.
4 = (4/3)π(3(2^2))(dr/dt)
Simplifying the equation:
4 = (4/3)π(12)(dr/dt)
4 = 16π(dr/dt)
Now, isolate dr/dt:
(dr/dt) = 4 / (16π)
(dr/dt) = 1 / (4π)
Therefore, when the radius is 2 feet, the rate at which the radius is increasing is 1 / (4π) feet per minute.
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 of the balloon and r is the radius. We need to differentiate this equation with respect to time t to find the rate at which the volume is changing over time.
dV/dt = d/dt[(4/3)πr^3]
Now, let's find the derivative of the equation with respect to t.
dV/dt = (4/3)π * d/dt[r^3]
Now, apply the chain rule to find d/dt[r^3].
dV/dt = (4/3)π * 3r^2 * dr/dt
We know that the balloon is being filled at a rate of 4 ft^3/min, meaning dV/dt = 4 ft^3/min. The radius is given as 2 feet, so r = 2 ft. We need to find dr/dt (the rate at which the radius is increasing) when r = 2 ft.
Substituting the given values into the equation:
4 = (4/3)π * 3(2)^2 * dr/dt
Now, solve for dr/dt:
dr/dt = 4 / [(4/3)π * 12]
Simplifying:
dr/dt = 4 / (16π)
So, the rate at which the radius is increasing when the radius is 2 feet is 4 /( 16π) ft/min.
you mean cubic feet per minute
surface area of sphere = 4 pi r^2
Well to avoid calculus just say the if you increase the radius by x, the volume will change by the outside surface area times that increase in radius x
so
increase in volume = 4 pi r^2 * increase in radius (that is the volume of the added skin of thickness x)
so
rate of change of volume = 4 pi r^2 * rate of increase of radius