A spherical snowball is melting in such a way that its diameter is decreasing at rate of 0.4 cm/min. At what rate is the volume of the snowball decreasing when the diameter is 10 cm. (The answer is a positive number).
To find the rate at which the volume of the snowball is decreasing, we need to use the formula for the volume of a sphere:
V = (4/3)πr^3,
where V is the volume and r is the radius of the sphere.
Since the problem provides the rate at which the diameter is decreasing, we need to find the rate at which the radius is decreasing. The radius is half of the diameter, so the rate at which the radius is decreasing is half the rate at which the diameter is decreasing.
Let's denote the rate at which the radius is decreasing as dr/dt. Since the problem gives the rate at which the diameter is decreasing as -0.4 cm/min (negative because it is decreasing), we have:
dr/dt = (-0.4 cm/min) / 2.
Substituting this value into the formula for the volume of a sphere, we have:
V = (4/3)πr^3,
dV/dt = (4/3)π(3r^2)(dr/dt) - [1],
where dV/dt represents the rate at which the volume is changing with respect to time.
We can now substitute the given diameter of 10 cm into equation [1] to find the rate at which the volume is decreasing:
r = 10 cm / 2 = 5 cm,
dr/dt = (-0.4 cm/min) / 2 = -0.2 cm/min.
Substituting these values into equation [1], we have:
dV/dt = (4/3)π(3(5^2))(-0.2).
= (4/3)π(75)(-0.2).
= -20π.
Therefore, the volume of the snowball is decreasing at a rate of 20π cubic centimeters per minute when the diameter is 10 cm.