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 16 cm. (Note the answer is a positive number).
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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) * π * r^3
where V is the volume and r is the radius of the sphere. We are given that the diameter is decreasing at a rate of 0.4 cm/min. Since the diameter is twice the radius, the rate at which the radius is decreasing is half the rate at which the diameter is decreasing.
Let's denote the radius as r and the rate of decrease as dr/dt. The given rate of decrease is -0.4 cm/min (negative because the radius is decreasing). So dr/dt = -0.4/2 = -0.2 cm/min.
Now we can differentiate the volume formula with respect to time (t) using the chain rule:
dV/dt = (4/3) * π * (3r^2)*(dr/dt)
Since we need to find the rate at which the volume is decreasing when the diameter is 16 cm (which corresponds to a radius of 8 cm), we substitute r = 8 into the equation:
dV/dt = (4/3) * π * (3(8^2)) * (-0.2)
dV/dt = (4/3) * π * 3 * 64 * -0.2
dV/dt = (4/3) * π * 192 * -0.2
dV/dt = -256π
Therefore, the volume of the snowball is decreasing at a rate of -256π cm^3/min, or approximately -804.248 cm^3/min (rounded to three decimal places).
Note that the answer is negative because the volume is decreasing, but the positive magnitude is |256π| or approximately 804.248 cm^3/min.