The surface area of a snowball can be represented by the formula:
A = 4πr^2
where A is the surface area, r is the radius of the snowball, and π is a constant approximately equal to 3.14.
If the surface area of the snowball decreases at a rate of 4 cm²/min, we can represent this as a derivative:
dA/dt = -4
where dA/dt is the rate of change of the surface area with respect to time, and t is the time in minutes.
The radius of the snowball is equal to half the diameter, so we can represent the radius as:
r = d/2
where d is the diameter of the snowball.
Substituting this expression for r into the formula for the surface area and differentiating with respect to time, we get:
dA/dt = 8π(d/2)dr/dt
Setting this expression equal to -4 and solving for dr/dt, we get:
dr/dt = -4 / (8π)
= -0.5 / π
When the diameter of the snowball is 39 cm, the rate at which the diameter decreases is:
dr/dt = -0.5 / π
= -0.15915494309189533576888376337251
Therefore, the rate at which the diameter of the snowball decreases when the diameter is 39 cm is approximately -0.159 cm/min.
I hope this helps