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 square foot, determine the dimensions for a box to have volume = 25 cubic feet which would minimize the cost of construction.
Values for dimension of the base are: ?
The height of the box is: ?
To determine the dimensions of the base, we can use the given information that the length of the base is 4 times larger than the width of the base. Let's assign a variable to the width of the base.
Let's say the width of the base is x.
Then the length of the base would be 4x.
To find the volume of the box, we can use the formula:
Volume = Length * Width * Height
We are given that the volume is 25 cubic feet, so we have:
25 = (4x) * x * Height
Multiplying the terms on the right side, we have:
25 = 4x^2 * Height
Now, let's solve for the height. Divide both sides by 4x^2:
25 / 4x^2 = Height
Simplifying, we have:
Height = 25 / (4x^2)
To minimize the cost of construction, we need to minimize the total surface area. The surface area of the base is given by:
Area of base = Length * Width = (4x) * x
The cost of constructing the base is $5 per square foot, so the cost of constructing the base is:
Cost of base = Area of base * $5 = (4x) * x * $5 = 20x^2
The surface area of the four sides of the box is given by:
Area of sides = (2 * Length * Height) + (2 * Width * Height) = (2 * (4x) * Height) + (2 * x * Height) = 8x * Height + 2x * Height = 10x * Height
The cost of constructing the four sides is $3 per square foot, so the cost of constructing the sides is:
Cost of sides = Area of sides * $3 = 10x * Height * $3 = 30x * Height
The total cost of construction is the sum of the cost of the base and the cost of the sides:
Total cost = Cost of base + Cost of sides = 20x^2 + 30x * Height
To minimize the total cost, we need to minimize this expression.
Substituting the expression for Height we found earlier, we have:
Total cost = 20x^2 + 30x * (25 / (4x^2))
Simplifying the expression further, we have:
Total cost = 20x^2 + (1875 / x)
To minimize the total cost, we can take the derivative of this expression with respect to x, set it equal to zero, and solve for x. Then we can substitute this value of x back into the equation to find the corresponding value of Height.
Taking the derivative, we have:
d(Total cost) / dx = 40x - (1875 / x^2)
Setting this equal to zero, we have:
40x - (1875 / x^2) = 0
Multiplying both sides by x^2, we have:
40x^3 - 1875 = 0
Simplifying further, we have:
40x^3 = 1875
Dividing both sides by 40, we have:
x^3 = 1875 / 40
Simplifying, we have:
x^3 = 46.875
Taking the cube root of both sides, we have:
x = cube root of (46.875)
Using a calculator, we find that the cube root of 46.875 is approximately 3.5707.
So the width of the base is approximately 3.5707 feet.
Since the length of the base is 4 times larger than the width, the length is approximately 4 * 3.5707 = 14.2828 feet.
To find the height, we can substitute these values of x and Height into the equation we found earlier:
Height = 25 / (4x^2) = 25 / (4 * (3.5707)^2) = 0.4361 feet.
So the dimensions for the box to have a volume of 25 cubic feet which would minimize the cost of construction are:
Width of the base: approximately 3.5707 feet
Length of the base: approximately 14.2828 feet
Height: approximately 0.4361 feet.
To determine the dimensions of the box that will minimize the cost of construction, we need to consider the cost equation. The cost of constructing the base of the box can be calculated by multiplying the area of the base by the cost per square foot, which is $5. The cost of constructing the four sides of the box can be calculated by multiplying the surface area of the sides (which is the perimeter of the base multiplied by the height of the box) by the cost per square foot, which is $3.
Let's start by determining the dimensions of the base. We are given that the length of the base is 4 times larger than the width of the base. Let's represent the width as W. Therefore, the length of the base would be 4W.
To find the dimensions of the base that will minimize the cost of construction, we need to express the cost equation in terms of one variable. Let's use the width of the base (W) as our variable.
The cost of constructing the base is given by:
Cost of base = Area of base * Cost per square foot
Cost of base = W * 4W * $5
The cost of constructing the four sides is given by:
Cost of sides = Surface area of sides * Cost per square foot
Cost of sides = (2W + 2(4W)) * H * $3
Cost of sides = (2W + 8W) * H * $3
Cost of sides = 10W * H * $3
The total cost of construction is the sum of the cost of the base and the cost of the sides.
Total cost = Cost of base + Cost of sides
Total cost = 4W^2 * $5 + 10W * H * $3
Now, we can express the volume of the box in terms of the width of the base (W) and the height of the box (H). Since the volume is given as 25 cubic feet, we have:
Volume = Length * Width * Height
25 = 4W * W * H
We now have two equations:
Total cost = 4W^2 * $5 + 10W * H * $3
25 = 4W * W * H
To find the dimensions that minimize the cost, we need to optimize the cost function subject to the volume constraint. We can use calculus to do this.
Differentiating the total cost equation with respect to W, we get:
d(Total cost)/dW = 8W * $5 + 10H * $3 = 40W + 30H
Setting the derivative equal to zero to find the minimum:
40W + 30H = 0
40W = -30H
4W = -3H
From the volume equation, we have:
W * W * H = 25
Substituting 4W = -3H from the first equation into the second equation, we get:
(-3/4W) * W * H = 25
-3/4W^2 = 25
W^2 = -25 * (4/3)
W^2 = -100/3
Since the width cannot be negative, there is no real solution to this equation. Therefore, there is no minimum that satisfies the given volume constraint.
Hence, there are no specific values for the dimensions of the base and the height of the box that will minimize the cost of construction while having a volume of 25 cubic feet.
l=5w
costbase= lw*5
cost sides=3(2lh+2wh)
volume=lwh
25=lwh
or h= 25/wl
costtotal=5(lw*5)+3(2l*25/wl + 2w*25/wl)
costtotal=25lw+ 150/w+150/l
but l=5w
costtotal= 125w^2+150/w+30/w
take the derivative of cost.
costtotal'=0=250w-150/w^2-30/w^2
solve for w:
250w^3-180=0
w^3=180/250
w= cubroot 7.2
l= 4 cubroot 7.2
h= 25/(lw)
check my work.