Posted by **john** on Monday, December 12, 2011 at 1:09am.

A tank with a rectangular base and rectangular sides is to be open at the top. It is to be constructed so that its width is 4 meters and its volume is 36 cubic meters. Building the tank costs $10 per square meter for the base and $5 per square meter for the sides.

a) Write a function C, the cost of constructing the described tank as a function of l(the length) and h(the height).

b) Write a function C, the cost of constructing the described tank as a function of a single variable.

c) What is the cost of the least expensive tank? (show all work)

- calculus -
**Reiny**, Monday, December 12, 2011 at 8:59am
let its length be l m

and its height by h m

volume = 4lh

but 4lh = 36

h = 9/l

a) cost = 10(4l) + 5(2lh) + 5(8h)

= 40l + 10lh + 40h

b) cost = 10(4l) + 5(2l)(9/l) + 40(9/l

= 40l + 90 + 360/l ,where l ≠ 0

c) d(cost)/dl = 40 + 0 - 360/l^2 = 0 for a min of cost

40 = 360/l^2

l^2 = 9

l = √9 = 3

when l = 3, h = 9/3 = 3

dimensions for min cost = 3by4 for the base and a height of 3

cost = 330.00

test: take a value slightly higher and lower than l = 3

l = 3.1, cost = 330.13 , higher

l = 2.9 , cost = 330.14 , higher

answer looks good

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