contractor wants to bid on an order to make 100,000 boxes from cardboard that costs $0.08 per square foot. Each box must have a square base and volume of 4 cubic feet. The sides of the box will be made from a single piece of cardboard, folded three times and taped at the seam. The top and bottom will be made from separate pieces of cardboard with the bottom taped on all four sides and the top taped on just one edge to create a lid that opens. Taping costs $0.03 per foot.
a) Find the dimensions of the box of minimum cost.
b) How much should the contractor bid to make a profit of 17%?
c) How much should be bid in question 2 be increased if the customer insists on cubical boxes?
could you explain me thoroughly please? thank you!!!
and show all your work lol so atleast i can follow it
5 stars 10 points. thanks!!
a) Find the dimensions of the box of minimum cost.
b) How much should the contractor bid to make a profit of 17%?
c) How much should be bid in question 2 be increased if the customer insists on cubical boxes?
could you explain me thoroughly please? thank you!!!
and show all your work lol so atleast i can follow it
5 stars 10 points. thanks!!
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The sides of the base are each x units. The height is y units.
A)
The volume is 4 cubic feet: y*x^2 = 4
Cost of material for the sides, top, and bottom: (2x^2 + 4xy)*.08
Cost of taping the base, one vertical edge, and one edge of the top: (4x + y + x)*.03
Total cost of the box, taping plus material: (2x^2 + 4xy)*.08 + (5x+y)*.03.
Additional requirement: x^2 * y = 4
Solve x^2*y = 4 for y and substitute in the cost expression.
x^2 * y = 4
y = 4/x^2
(2x^2 + 4xy)*.08 + (5x+y)*.03
Substitute for y
(2x^2 + 4x*4/x^2)*.08 + (5x + 4/x^2)*.03
(2x^2 + 16/x)*.08 + (5x + 4/x^2)*.03
.16x^2 + 1.28/x + .15x + .12/x^2
So, the total cost in terms of x is .16x^2 + 1.28/x + .15x + .12/x^2 and you have to find the x which gives a minimum value. (Find where the derivative is zero. See the links below.) There are two places where a local minimum occurs but one of those is a negative x, so it is invalid. The other local minimum occurs when x is approximately 1.50824 and the cost is approximately $1.49163 per box.
y = 4/x^2 which is approximately 1.75841
Minimum cost is a box with base 1.50824 feet on each side and height 1.75841 feet.
The cost of that box is $1.49163.
B)
To make a 17% profit if x is the sale price per box then x/1.49163 = 1.17.
x = 1.49163 * 1.17 = 1.7452071
So the price should be $1.7452071 per box.
For 100,000 boxes the bid should $1,745,207.10
C)
For cubical boxes we have
x^3 = 4
Taping costs = 6x*.03
Material cost = 6x^2*.08
Total cost = .48*x^2 + .18*x
x = 4^(1/3)
Total cost is .48 * 4(^2/3) + .18 * 4^(1/3) = approximately 1.4952564.
To get a 17% profit the sale price should be 1.4952564*1.17 or 1.74944999.
For 100,000 boxes the bid should be $1,749,449.99.
That is more than the original bid by $1,749,449.99 - $1,745,207.10 = $4,242.89.
The sides of the base are each x units. The height is y units.
A)
The volume is 4 cubic feet: y*x^2 = 4
Cost of material for the sides, top, and bottom: (2x^2 + 4xy)*.08
Cost of taping the base, one vertical edge, and one edge of the top: (4x + y + x)*.03
Total cost of the box, taping plus material: (2x^2 + 4xy)*.08 + (5x+y)*.03.
Additional requirement: x^2 * y = 4
Solve x^2*y = 4 for y and substitute in the cost expression.
x^2 * y = 4
y = 4/x^2
(2x^2 + 4xy)*.08 + (5x+y)*.03
Substitute for y
(2x^2 + 4x*4/x^2)*.08 + (5x + 4/x^2)*.03
(2x^2 + 16/x)*.08 + (5x + 4/x^2)*.03
.16x^2 + 1.28/x + .15x + .12/x^2
So, the total cost in terms of x is .16x^2 + 1.28/x + .15x + .12/x^2 and you have to find the x which gives a minimum value. (Find where the derivative is zero. See the links below.) There are two places where a local minimum occurs but one of those is a negative x, so it is invalid. The other local minimum occurs when x is approximately 1.50824 and the cost is approximately $1.49163 per box.
y = 4/x^2 which is approximately 1.75841
Minimum cost is a box with base 1.50824 feet on each side and height 1.75841 feet.
The cost of that box is $1.49163.
B)
To make a 17% profit if x is the sale price per box then x/1.49163 = 1.17.
x = 1.49163 * 1.17 = 1.7452071
So the price should be $1.7452071 per box.
For 100,000 boxes the bid should $1,745,207.10
C)
For cubical boxes we have
x^3 = 4
Taping costs = 6x*.03
Material cost = 6x^2*.08
Total cost = .48*x^2 + .18*x
x = 4^(1/3)
Total cost is .48 * 4(^2/3) + .18 * 4^(1/3) = approximately 1.4952564.
To get a 17% profit the sale price should be 1.4952564*1.17 or 1.74944999.
For 100,000 boxes the bid should be $1,749,449.99.
That is more than the original bid by $1,749,449.99 - $1,745,207.10 = $4,242.89.