An open box with a capacity of 36,000 cubic inches is to be twice as long as it is wide. The material for the box is $.10/sq ft. What are the dimensions of the least expensive box? How much does it cost?
Since the cost is the same for all surface areas, we just want to find the box with minimum surface area that holds the given volume.
The surface area involved is one base and two short sides and two long sides, so
a = 2x^2 + 2xz + 4xz = 2x^2+6xz
But since z = 36000/2x^2 = 18000/x^2,
a(x) = 2x^2 + 108000/x
da/dx = 4x - 108000/x^2
= (4x^3-108000)/x^2
= 4(x^3-27000)/x^2
clearly, da/dx = 0 when x=30
Now you can just provide the dimensions and the cost.
To find the dimensions of the least expensive box, we need to minimize the cost of the material used.
Let's assume the width of the box is x inches. Since the length is twice the width, the length would be 2x inches.
The height of the box is not mentioned, so we can assume it to be h inches.
The volume of the box can be calculated as follows:
Volume = Length x Width x Height
Volume = (2x) x (x) x h
Given that the volume is 36,000 cubic inches, we have:
36,000 = 2x^2h
The cost of the material is given as $0.10/sq ft. To calculate the cost, we need to find the surface area of the box.
The surface area of the box can be calculated as follows:
Surface Area = 2(length x width) + 2(length x height) + 2(width x height)
Substituting the values, we get:
Surface Area = 2(2x^2) + 2(2xh) + 2(xh)
Surface Area = 4x^2 + 4xh + 2xh
Now, let's solve the equations to find the dimensions and cost.
1. Substitute the equation for volume into the surface area equation:
Surface Area = 4x^2 + 4xh + 2xh = 4x^2 + 6xh
2. Substitute x from the equation for volume:
Surface Area = 4(36,000/2h) + 6(36,000/h)
Surface Area = 72,000/h + 216,000/h
Surface Area = (72,000 + 216,000)/h
Surface Area = 288,000/h
3. To minimize cost, we need to minimize the surface area:
To minimize the value of the surface area, we need to maximize the value of h.
To find the optimal value of h, we take the derivative of the surface area equation with respect to h and set it equal to zero:
d(Surface Area)/dh = -288,000/h^2 = 0
Solving for h, we get:
h^2 = 288,000
h = sqrt(288,000)
h ≈ 537.41 inches
4. Substitute the value of h back into the equation for x:
36,000 = 2x^2(537.41)
x^2 = 36,000 / (2 * 537.41)
x^2 ≈ 33.38
x ≈ sqrt(33.38)
x ≈ 5.78 inches (approx.)
Therefore, the dimensions of the least expensive box are approximately:
Width ≈ 5.78 inches
Length ≈ 11.56 inches
Height ≈ 537.41 inches
To find the cost, we need to calculate the surface area and multiply it by the cost per square inch:
Surface Area = 288,000 / 537.41 ≈ 536.96 square inches
The cost of the material is $0.10/sq ft, which is equal to $0.10 / (12 x 12) = $0.00694 per square inch.
Total Cost = Surface Area x Cost per square inch
Total Cost ≈ 536.96 x $0.00694
Total Cost ≈ $3.73
Therefore, the cost of the material for the least expensive box is approximately $3.73.
To find the dimensions of the least expensive box, we need to consider both the volume of the box and the material cost. Let's break down the problem step by step:
Step 1: Determine the dimensions of the box.
Let's assume the width of the box is x inches. Given that the length is twice the width, the length of the box is 2x inches.
Step 2: Calculate the volume of the box.
The formula for the volume of a rectangular box is V = length x width x height. However, since this is an open box, there is no height. So, the formula becomes V = length x width.
Given that the capacity of the box is 36,000 cubic inches, we can write the equation as:
36,000 = (2x) * x.
Step 3: Solve for x.
36,000 = 2x^2.
Divide both sides by 2:
18,000 = x^2.
Take the square root of both sides:
x = √18,000.
The exact value of the square root may be a bit complex, but we can approximate it. Rounded to the nearest whole number, x ≈ 134.
Step 4: Calculate the dimensions of the box.
The width is approximately 134 inches, and the length is twice the width, so the length is approximately 2 * 134 = 268 inches.
Step 5: Calculate the cost.
To find the cost, we need to determine the surface area of the box. The surface area of a rectangular box is given by the formula:
A = 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height.
Since this is an open box, we don't have a height.
Therefore, the formula simplifies to:
A = 2lw + 2wh.
Given that the material cost is $0.10 per sq ft, we need to convert the surface area into sq ft:
Surface Area (sq ft) = (2lw + 2wh) / 12^2.
Substituting the values:
Surface Area (sq ft) = (2 * 268 * 134 + 2 * 268 * 134) / 12^2.
Calculating it further, the surface area (approximately) ≈ 649.78 sq ft.
Finally, to calculate the cost, multiply the surface area by the cost per sq ft:
Cost = Surface Area * $0.10.
Cost (approximately) ≈ 649.78 sq ft * $0.10 = $64.98.
Therefore, the dimensions of the least expensive box are approximately 134 inches by 268 inches, and it costs approximately $64.98.