Answer:
Therefore, when the diameter is 10 cm, the rate at which the diameter decreases is approximately -0.572 cm/min.
Explanation:
To find the rate at which the diameter of the snowball decreases, we need to relate the surface area and the diameter of the snowball.
The surface area of a sphere is given by the formula:
A = 4πr^2
where A is the surface area and r is the radius of the sphere.
Since the diameter (d) is twice the radius (r), we can write the formula for surface area in terms of the diameter as:
A = π(d/2)^2
A = (π/4)d^2
We are given that the surface area is decreasing at a rate of 9 cm^2/min. So, we can express this rate of change as:
dA/dt = -9
where dA/dt represents the rate of change of surface area with respect to time (t).
To find the rate at which the diameter (d) decreases when the diameter is 10 cm, we need to find dd/dt (rate of change of the diameter with respect to time) when d = 10.
First, differentiate the equation for the surface area with respect to time:
dA/dt = (π/4)(2d)(dd/dt)
-9 = (π/2)(10)(dd/dt)
-9 = 5π(dd/dt)
Now, solve for dd/dt:
dd/dt = (-9)/(5π)
Using a calculator, this simplifies to approximately -0.572 cm/min.
Therefore, when the diameter is 10 cm, the rate at which the diameter decreases is approximately -0.572 cm/min.