The total surface area for a silo (base with radius r, cylindrical side with height h, hemispherical cap) is A = 3πr2 + 2πrh. Suppose h and r are measured to be h = 20 feet and r = 8 feet to within an error of 1 2 foot for h and 1 4 foot for r. Use differentials to estimate the maximum possible error for A (in square feet).

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Respuesta :

Manetho Manetho · Answer 1 · 2019-12-12T12:09:25+01:00

Answer:

[tex]\Delta A=30\pi ft^2[/tex]

Step-by-step explanation:

The total surface area for a silo A = 3πr^2 + 2πrh.

given

base radius r, cylindrical side with height h, hemispherical cap

h=20 ft r=8 ft

Δh= 1/2 ft and Δr= 1/4 ft

now, diffrentiating we get

dA= 3π×2r×dr+2π(r×dh+Δr×h)

putting values we get

[tex]\Delta A= 6\pi\Delta r+2\pi r\Delta h+2\pi h \Delta r[/tex]

[tex]\Delta A= (6\pi\ r+2\pi h)\Delta r+2\pi r \Delta h[/tex]

[tex]\Delta A= (6\pi\ 8+2\pi 20)\(1/4)+2\pi 8(1/2)[/tex]

solving the above equation we get

[tex]\Delta A=30\pi[/tex]