calculus

posted by .

A box is constructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $2/ft2. The metal for the four rectangular sides costs $3/ft2. Find the dimensions that minimize cost if the box has a volume 20 ft3.

Top and bottom should be squares with sides of length L = ft
The height of the box should be h = ft

  • calculus -

    Cost = cost of top + cost of bottom + cost of 4 sides
    = 2L^2 + 2L^2 + 3(4Lh)
    = 4L^2 + 12Lh

    but L^2 h = 20
    h = 20/L^2

    so Cost = 4L^2 + 12L(20/L^2)
    = 4L^2 + 240/L
    d(Cost)/dL = 8L - 240/L^2 = 0 for max/min of Cost

    8L = 240/L^2
    L^3 = 30
    L = 3.107

    then h = 20/3.107^2 = 2.07

  • calculus -

    V-Volume
    V=20
    V=L^2*h
    L^2*h=20 Divide with (L^2)
    h=20/(L^2)
    Area of square is L^2
    Areao of rectacangle L*h
    Total price=2*Area of square*(2$)+4*Area of rectacangle*(3$)
    P-Price
    P=2*(L^2)*2+4*L*h*3
    =4*L^2+12*L*20/(L^2)
    =4*L^2+240*L/(L^2)
    =4*L^2+240/L

    (dP/dL)=4*2*L+240*(-1)/(L^2)
    (dP/dL)=8*L-240/(L^2)
    Price have minimum where is (dP/dL)=0

    8*L-240/(L^2)=0
    8*L=240/(L^2) Divide with 8
    L=30/(L^2) Multiply with L^2
    L^3=30

    L=third root of 30

    h=20/(L^2)
    =20/(third root of 30)^2

    h=20/third root of 900

    Proof that is minimum
    Function have minimum when:

    (dP/dL)=0 and (d^2P)/dL^2>0

    First derivation=0 and
    second derivation higher of zero

    (d^2P)/dL^2=Derivation of first derivation

    (d^2P)/dL^2=8-240*(-2)/L^3
    =8+480/L^3

    8+480/L^3 is always higher of zero.

    Function have minimum.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. calculus

    A cylindrical metal can with an open top is to be constructed. the can must have a capacity of 24pie cubic inches. the metal used to contruct the bottom of the can costs 3 times as much as the metal in the rest of the can. Find the …
  2. calculus

    A rectangular box is to have a square base and a volume of 50 ft3. The material for the base costs 44¢/ft2, the material for the sides costs 10¢/ft2, and the material for the top costs 26¢/ft2. Letting x denote the length of one …
  3. calculus

    A rectangular box is to have a square base and a volume of 50 ft3. The material for the base costs 44¢/ft2, the material for the sides costs 10¢/ft2, and the material for the top costs 26¢/ft2. Letting x denote the length of one …
  4. calculus

    A box is constructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $4 per square foot and the metal for the sides costs $4 per square foot. Find the dimensions that minimize cost …
  5. Calculus

    A box is constructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $2 per square foot and the metal for the sides costs $7 per square foot. Find the dimensions that minimize cost …
  6. Calculus

    A rectangular box is to have a square base and a volume of 20 ft3. The material for the base costs 42¢/ft2, the material for the sides costs 10¢/ft2, and the material for the top costs 26¢/ft2. Letting x denote the length of one …
  7. calculus

    A box is contructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $5 per square foot and the metal for the sides costs $2 per square foot. Find the dimensions that minimize cost …
  8. Calculus

    Gloria would like to construct a box with volume of exactly 45ft^3 using only metal and wood. The metal costs $15/ft^2 and the wood costs $6/ft^2. If the wood is to go on the sides, the metal is to go on the top and bottom, and if …
  9. Calc

    Gloria would like to construct a box with volume of exactly 55ft3 using only metal and wood. The metal costs $12/ft2 and the wood costs $9/ft2. If the wood is to go on the sides, the metal is to go on the top and bottom, and if the …
  10. Calculus

    Suppose that you are to make a rectangular box with a square base from two different materials. The material for the top and four sides of the box costs $1/ft2$1/ft2; the material for the base costs $2/ft2$2/ft2. Find the dimensions …

More Similar Questions