Show all steps (all the necessary algebra) in your work. Simply solving the prob

Question

Show all steps (all the necessary algebra) in your work. Simply solving the problem with Wolfram or similar software will not give you any credit for these problems. In sixteenth Austria, wine barrels were measured by how far a bung stick (labeled TP in figure) could be inserted through the bunghole in the middle of the side of a barrel. Winemaker Beavis calculates the radius and height of a cylindrical wine barrel with volume 8 pi cubic feet that has the smallest bung stick measurement What values does he obtain?

Explanation / Answer

let the radius of the drum be r
let its height be h
then pi*r^2h = 8*pi
hr^2 = 8 .. (1)

now, the bunghople is at the middle of the side of the barrel
so, if length of the bungstick inside the hole is l
then l^2 = (2r)^2 + (h/2)^2 [ from pythagoras theorem]
for l to be minimum, l^2 is minimum as well
do d(l^2)/dh = 0 ( using conditions for maxima and minima)

so l^2 = 4r^2 + h^2/4 = 32/h + h^2/4
dl^2/dh = 0 = 2h/4 - 32/h^2 = 0
h/2 = 32/h^2
h^3 = 32*2 = 64
h = 4

so, r = sqroot(2)

hence length of the stick inside = l = sqroot(4r^2 + h^2/4) = sqroot(4*2 + 4^2/4) = sqroot(8 + 4) = sqroot(12) = 3.464 ft

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