The radius of a sphere is increasing at a rate of 2 mm/s. How fast is the volume increasing when the diameter is 100 mm? Evaluate your answer numerically.
To find the rate at which the volume of a sphere is increasing, we need to use the mathematical relationship between the radius and the volume.
The formula for the volume of a sphere is given by:
V = (4/3) * π * r^3,
where V is the volume and r is the radius.
Since we are given that the radius is increasing at a rate of 2 mm/s, we can express this as dr/dt = 2 mm/s, where dr/dt is the rate of change of the radius with respect to time.
We are also given that the diameter is 100 mm. Since the diameter is twice the radius, we can calculate the initial radius as r = d/2 = 100 mm / 2 = 50 mm.
Now, let's differentiate the volume formula with respect to time t to find the rate of change of the volume with respect to time:
dV/dt = d/dt [(4/3) * π * r^3].
Using the chain rule of differentiation, we can write:
dV/dt = (4/3) * π * d/dt (r^3).
To find dV/dt, we only need to find the value of d/dt (r^3).
Taking the derivative of r^3 with respect to t gives us:
d/dt (r^3) = 3r^2 * (dr/dt).
Substituting the given values, we can calculate the rate at which the volume is increasing:
dV/dt = (4/3) * π * (50 mm)^2 * (2 mm/s).
= (4/3) * π * 2500 mm^2 * (2 mm/s).
Evaluating this numerically gives us:
dV/dt ≈ 16728 mm^3/s.
v=4/3 pi r^3
dv/dt= 4PI r^2 dr/dt
dv/dt= 4*PI*(50)^2 * 2 mm^3/sec