Calc Problem for Extra Credit!?
A 100ft cube is resting with one face on the ground. The remaining faces are painted. If the coating of paint is a uniform 0.0024 inches in thickness, approximate the volume of the paint used. Find the actual amount of paint used? How good is the approximation?
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Verified answer
Let:
x be the side of the cube,
S be the surface area of the five painted surfaces,
V be the volume of the cube.
S = 5x^2
V = x^3
dV / dx = 5 * 3x^2 / 6
Approximately:
delta V = 2.5 x^2 delta x
Putting x = 100ft, delta x = 0.024in = 0.002ft:
delta V = 2.5 * 100^2 * 0.0002
delta V = 5ft^3.
Exactly:
delta V = 5(100.0002^3 - 100^3) / 6
= 5.000009... ft^3
Accuracy of the approximation is better than 1 in 500,000.