diffirential calculus

posted by .

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 -

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
V= (3x)(4x)(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)

Similar Questions

1. CALC

An open box is to be constructed so that the length of the base is 4 times larger than the width of the base. If the cost to construct the base is 5 dollars per square foot and the cost to construct the four sides is 3 dollars per …
2. calculus

An open box is to be constructed so that the length of the base is 4 times larger than the width of the base. If the cost to construct the base is 5 dollars per square foot and the cost to construct the four sides is 3 dollars per …
3. Calculus optimization

A rectangular storage container with a lid is to have a volume of 8 m. The length of its base is twice the width. Material for the base costs \$4 per m. Material for the sides and lid costs \$8 per m. Find the dimensions of the container …
4. HELP!! OPTIMIZATION CALCULUS

A rectangular storage container with a lid is to have a volume of 8 m. The length of its base is twice the width. Material for the base costs \$4 per m. Material for the sides and lid costs \$8 per m. Find the dimensions of the container …
5. Calculus

AM having problems understanding what equations to use for this word problem. Please help. A rectangular storage container with an open top is to have a volume of 48ft^3. The length of its base is twice the width. Material for the …
6. Calculus

A rectangular storage container with an open top is to have a volume of 10m^3. The length of its base is twice the width. Material for the base costs \$3 per m^2. Material for the sides costs \$10.8 per m^2. Find the dimensions of the …
7. Calculus

A holding pen for fish is to be made in the form of a rectangular solid with a square base and open top. The base will be slate that costs \$4 per square foot and the sides will be glass that costs \$5 per square foot. If the volume …
8. Calc 1

A rectangular storage container with an open top is to have a volume of 10 m3. The length of the base is twice the width. Material for the base is thicker and costs \$13 per square meter and the material for the sides costs \$10 per …
9. Math

A toolshed with a square base and a flat roof is to have a volume of 800 cubic feet. If the floor costs \$6 per square foot, the roof \$2 per square foot, and the sides \$5 per square foot, determine the dimensions of the least expensive …
10. Calculus

A rectangular storage container with an open top is to have a volume of 10 . The length of its base is twice the width. Material for the base costs \$12 per square meter. Material for the sides costs \$5 per square meter. Find the cost …

More Similar Questions