Posted by kait on Sunday, September 25, 2011 at 12:26am.
a builder intends to construct a storage shed having a volume of 900ft^3, a flat roof and a rectangular base whose width is three-fourths the length. the cost per square foot of the materials is 4000.00 for the floor,6000.00 for the sides and 3000.00 for the roof. what dimension will minimize the cost?
- diffirential calculus - Reiny, Sunday, September 25, 2011 at 11:18am
I like to avoid fractions if possible, so let the
width be 3x
and the length be 4x , (notice that 3x/4x) = 3/4)
let the height be h
h = 900/(12x^2) = 75/x^2
Cost = C = 4000(top) + 6000(sides) + 3000(roof)
= 4000(12x^2) + 6000(8xh+6xh) + 3000(12x^2)
= 84000x^2 + 84000xh
= 84000x^2 + 84000x(75/x^2
= 84000x^2 + 84000(75)/x
= 84000(x^2 + 75/x)
d(Cost)/dx = 84000(2x - 75/x^2) = 0 for a min of C
2x = 75/x^2
x^3 = 75/2 = 37.5
x = (37.5)^(1/3) = 3.347
width = 3(3.347) = 10.04
length = 4(3.347) = 13.39
height = 75/(3.347)^2 = 6.69 (all in feet)
Answer This Question
More Related Questions
- Calculus - AM having problems understanding what equations to use for this word...
- Math - A toolshed with a square base and a flat roof is to have a volume of 800 ...
- calculus - An open box is to be constructed so that the length of the base is 4 ...
- CALC - An open box is to be constructed so that the length of the base is 4 ...
- Calculus - A rectangular storage container with an open top is to have a volume ...
- Calc 1 - A rectangular storage container with an open top is to have a volume of...
- Calculus - A holding pen for fish is to be made in the form of a rectangular ...
- HELP!! OPTIMIZATION CALCULUS - A rectangular storage container with a lid is to...
- Calculus optimization - A rectangular storage container with a lid is to have a ...
- Calc 1 (Optomization) - A rectangular storage container with an open top is to ...