A globe with a diameter of 30 cm is to be painted with a uniform layer of paint which is 4 mm thick. Use the method of differentiation to find the approximate volume of paint needed.
dV = 4pi(15^2)(0.4) = 360pi cu.cm.
Not the 160pi cu.cm I wrote before.
But the answer should be 1440pi cu.cm?
Let us check by geometry.
>>>sphere with no paint, r = 15cm
V1 = (4/3)pi(15^3) = 4500pi cc.
>>>sphere painted 4mm thick, r = 15.4 cm
V2 = (4/3)pi(15.4^3) = 4869.7 cc
So, volume of paint = V2 -V1 = 4869.7pi -4500pi = 369.7pi
That is much closer to the 360pi cc than to the 1440pi cc.
Is this right? I drew a diagram and tried separating the paint into a cuboid by using 0.4 cm as the height, 30 pi (circumference) as the length, and dA over dr, 120 pi (surface area) as the depth. My answer is 1440 pi^2 cm cube which looks a lot like the answer, but I'm sure of otherwise. Help anyone?