A spherical snowball is melting in such a way that its diameter is decreasing at rate of 0.2 cm/min. At what rate is the volume of the snowball decreasing when the diameter is 12 cm. (Note the answer is a positive number).
V = (pi/6)D^3
-dV/dt = -3*(pi/6)*D^2*dD/dt
= -(pi/2)*D^2*dD/dt
The rate of decrease of diameter, dD/dt, is negative, so the answer is positive.
Now, perform the calculation.
45.2389342117
To find the rate at which the volume of the snowball is decreasing, we can use the formula for the volume of a sphere:
V = (4/3)πr^3,
where V is the volume and r is the radius.
We are given that the diameter, which is twice the radius, is decreasing at a rate of 0.2 cm/min. Let's call this rate of change dr/dt. Since dr/dt is a negative value (because the diameter is decreasing), we can rewrite it as:
dr/dt = -0.2 cm/min.
We need to find the rate at which the volume, dV/dt, is decreasing when the diameter is 12 cm. Since the diameter is twice the radius, when the diameter is 12 cm, the radius will be 6 cm.
To find dV/dt, we can differentiate the volume formula with respect to time (t):
dV/dt = d/dt [(4/3)πr^3].
Using the chain rule, we can write this as:
dV/dt = (4/3)π * d/dt (r^3).
To find d/dt (r^3), we can differentiate r^3 with respect to t:
d/dt (r^3) = 3r^2 * dr/dt.
Substituting the values we know:
dV/dt = (4/3)π * 3r^2 * dr/dt.
When the diameter is 12 cm, the radius is 6 cm, so we can substitute r = 6:
dV/dt = (4/3)π * 3(6)^2 * (-0.2).
Simplifying this:
dV/dt = (4/3)π * 3 * 36 * (-0.2).
dV/dt = (4/3)π * 3 * 36 * (-0.2).
dV/dt = (4π) * (-7.2).
dV/dt = -28.8π.
Therefore, the volume of the snowball is decreasing at a rate of -28.8π cubic cm per minute when the diameter is 12 cm.
Since the question asks for a positive value, we can ignore the negative sign. The rate at which the volume of the snowball is decreasing when the diameter is 12 cm is 28.8π cubic cm/min.