A company will make a cereal box with whole number dimensions and a volume of 100 cubic centimeters. If cardboard costs $0.05 per 100 square centimeters, what is the least cost to make 100 boxes?
Please help me, I have no clue how to solve.
To find the least cost to make 100 cereal boxes, we need to consider the dimensions of the cereal box that would result in the minimum cost, given that the volume of each box is 100 cubic centimeters.
Let's assume the dimensions of the cereal box are length, width, and height, all measured in centimeters.
Since the volume of a rectangular cuboid (such as the cereal box) is given by the formula length x width x height, we have:
length x width x height = 100 cubic centimeters ...(1)
Now, to minimize the cost, we need to minimize the surface area of the cereal box because we are using cardboard, which is priced based on its area. The cost is given as $0.05 per 100 square centimeters, which means for every 100 square centimeters of cardboard used, it costs $0.05.
The surface area of a rectangular cuboid can be calculated as follows:
2(length x width + width x height + height x length) ...(2)
Now, let's proceed to find the least cost by considering different dimensions for the cereal box.
1. We'll start by considering the dimensions where the length, width, and height are all equal (i.e., a cube).
Substituting these equal values into equation (1), we have:
length^3 = 100 cubic centimeters
Taking the cube root of both sides, we find:
length = ∛100 ≈ 4.64 cm
Now, substitute this length value into equation (2) to find the surface area:
2(length x length + length x length + length x length)
= 2(4.64 x 4.64 + 4.64 x 4.64 + 4.64 x 4.64)
≈ 2(21.57 + 21.57 + 21.57)
≈ 2(64.71) ≈ 129.42 square centimeters
To calculate the cost, we divide the surface area by 100 and multiply it by $0.05:
Total cost for 1 cube-shaped cereal box = (129.42 / 100) x $0.05 ≈ $0.06
Therefore, the total cost to make 100 cube-shaped cereal boxes would be 100 x $0.06 = $6.00.
2. Next, we'll consider the dimensions where the length is twice the width and the height is half the width.
Substituting these values into equation (1), we have:
(2w) x w x (0.5w) = 100 cubic centimeters
Simplifying, we find:
w^3 = 100 cubic centimeters
Taking the cube root of both sides, we find:
w = ∛100 ≈ 4.64 cm
Now, we can substitute this width value into the length and height equations to find the values for the respective dimensions:
length = 2w ≈ 2(4.64) = 9.28 cm
height = 0.5w ≈ 0.5(4.64) = 2.32 cm
Substituting these dimensions into equation (2), we have:
2((2w x w) + (w x 0.5w) + (0.5w x 2w))
= 2((9.28 x 4.64) + (4.64 x 2.32) + (2.32 x 9.28))
≈ 2(42.95 + 21.57 + 42.95)
≈ 2(107.47) ≈ 214.94 square centimeters
To calculate the cost, we divide the surface area by 100 and multiply it by $0.05:
Total cost for 1 cereal box with dimensions as described = (214.94 / 100) x $0.05 ≈ $0.11
Therefore, the total cost to make 100 cereal boxes with these dimensions would be 100 x $0.11 = $11.00.
After comparing the costs for both cases, we can conclude that the least cost to make 100 cereal boxes with a volume of 100 cubic centimeters is $6.00, using cube-shaped boxes with equal dimensions for length, width, and height.
Note: This solution assumes that the cereal boxes have an open top and do not include any flaps or additional material beyond the surface area calculations provided.