The weight of a concrete column

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Question from russell, a student:

a cylindrical form is filled with a slow curing concrete. The base of the form is 10 ft in radius, and height is 25 ft. while the concrete hardens, gravity causes the density to vary from a density of 90 lbs/ft^3 at the bottom to a density of 50 lb/ft^3 at the top. Assume that the density varies linearly
from the top to the bottom, and compute the total weight of the resulting concrete column

Hi Russell,

Suppose D(x) is the density of the concrete x feet up from the bottom. You know that D(x) is a linear function, D(0) = 90 lbs/ft³ and D(25) = 50 lbs/ft³. Find the linear function D(x).

Suppose that you have thin slab of the concrete, x feet up from the bottom and of thickness Δx feet. The slab is a cylinder so its volume is π r² h = π 10² Δx = 100 π Δx ft³. The density of the slab is D(x) lbs/ft³ so its weight is 100 π D(x) Δx.

Can you now complete the problem?

Since the density varies linearly it seems to me that the weight of the concrete column should be the volume of the column times the density half way up. Does this agree with your answer?

Harley

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