A hemispherical dome of radius 40 feet is to be given 7 coats of paint, each of which is 1/100 inch thick. How to use linear approximation to estimate the volume of paint needed for the job?

Layla Melton

Layla Melton

Answered question

2023-03-24

A hemispherical dome of radius 40 feet is to be given 7 coats of paint, each of which is 1/100 inch thick. How to use linear approximation to estimate the volume of paint needed for the job?

Answer & Explanation

Razorel1l1

Razorel1l1

Beginner2023-03-25Added 6 answers

I'd go with the conclusion that, for a sphere:
V = 4 3 π r 3 and d V d r = 4 π r 2
[ie the surface area of a sphere of radius r is the derivative wrt r of its volume]
To begin with, we can state that:
δ V = d V d r δ r = 4 π r 2 δ r
and so with δ r = 7 1 12 100 (7 layers, and adjusting to Imperial ft measurements), we have
δ V = 4 π ( 40 ) 2 7 100 = 112 3 π 117.3 ft 3
Actual increase is
Δ V = 4 3 π ( ( 40 + 7 1200 ) 3 - 40 3 ) = 117.3 ft 3

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