A company will make a cereal box with whole number dimensions and the volume of 100 cm³. If cardboard cost five cents per 100 cm² what is the least cost to make 100 boxes?
To find the least cost to make 100 cereal boxes, we need to determine the dimensions of the boxes that would result in the least amount of cardboard used.
Let's assume the dimensions of the cereal box are length (L), width (W), and height (H), all measured in cm. The volume of the cereal box is given as 100 cm³.
The volume of a rectangular box is calculated by multiplying its length, width, and height: Volume (V) = L x W x H.
In this case, V = 100 cm³. Since V needs to be a whole number, let's start by finding the factors of 100.
The factors of 100 are: 1, 2, 4, 5, 10, 20, 25, 50, and 100.
To minimize the cost of cardboard, we need to find the combination of L, W, and H that results in the smallest surface area. The surface area of a rectangular box is calculated by adding the areas of its individual faces.
The formula for the surface area (A) of a rectangular box is: A = 2(L x W + L x H + W x H).
Let's evaluate the surface areas for each combination of factors:
For L=1, W=4, and H=25, A = 2(1x4 + 1x25 + 4x25) = 234 cm²
For L=1, W=5, and H=20, A = 2(1x5 + 1x20 + 5x20) = 230 cm²
For L=2, W=2, and H=25, A = 2(2x2 + 2x25 + 2x25) = 216 cm²
For L=2, W=4, and H=10, A = 2(2x4 + 2x10 + 4x10) = 120 cm²
For L=4, W=5, and H=5, A = 2(4x5 + 4x5 + 5x5) = 106 cm²
Based on the above calculations, the combination of L=4, W=5, and H=5 produces the smallest surface area, resulting in the least amount of cardboard used.
The surface area required for one box is 106 cm². The surface area required for 100 boxes will be 100 x 106 cm² = 10,600 cm².
Given that cardboard costs five cents per 100 cm², the cost of the cardboard for 100 boxes will be (10,600 / 100) x 5 cents = $53.
Therefore, the least cost to make 100 boxes would be $53.
To find the least cost to make 100 cereal boxes, we need to determine the dimensions of the cereal box with a volume of 100 cm³ that will minimize the cost of cardboard.
Let's start with the volume equation for a rectangular box:
Volume = length × width × height
Given that the volume of the cereal box is 100 cm³ and the dimensions are whole numbers, we need to find the combinations of whole numbers that satisfy the equation.
Let's analyze the possible dimensions:
Since the volume is 100 cm³, the dimensions could be 1×1×100, 1×2×50, 1×4×25, 1×5×20, 1×10×10, 2×2×25, 2×5×10, or 4×5×5.
Now, let's calculate the surface area of each box:
Surface Area = 2lw + 2lh + 2wh
Using the dimensions mentioned above, we can calculate the surface area for each box:
- For 1×1×100: Surface Area = 2(1)(1) + 2(1)(100) + 2(1)(100) = 2 + 200 + 200 = 402 cm²
- For 1×2×50: Surface Area = 2(1)(2) + 2(1)(50) + 2(2)(50) = 4 + 100 + 200 = 304 cm²
- For 1×4×25: Surface Area = 2(1)(4) + 2(1)(25) + 2(4)(25) = 8 + 50 + 200 = 258 cm²
- For 1×5×20: Surface Area = 2(1)(5) + 2(1)(20) + 2(5)(20) = 10 + 40 + 200 = 250 cm²
- For 1×10×10: Surface Area = 2(1)(10) + 2(1)(10) + 2(10)(10) = 20 + 20 + 200 = 240 cm²
- For 2×2×25: Surface Area = 2(2)(2) + 2(2)(25) + 2(2)(25) = 16 + 100 + 100 = 216 cm²
- For 2×5×10: Surface Area = 2(2)(5) + 2(2)(10) + 2(5)(10) = 20 + 40 + 100 = 160 cm²
- For 4×5×5: Surface Area = 2(4)(5) + 2(4)(5) + 2(5)(5) = 40 + 40 + 50 = 130 cm²
Now that we have the surface area for each box, let's calculate the cost of cardboard for each combination. Given that cardboard costs five cents per 100 cm²:
- For 1×1×100: Cost = (402/100) × 0.05 = 2.01 cents
- For 1×2×50: Cost = (304/100) × 0.05 = 1.52 cents
- For 1×4×25: Cost = (258/100) × 0.05 = 1.29 cents
- For 1×5×20: Cost = (250/100) × 0.05 = 1.25 cents
- For 1×10×10: Cost = (240/100) × 0.05 = 1.2 cents
- For 2×2×25: Cost = (216/100) × 0.05 = 1.08 cents
- For 2×5×10: Cost = (160/100) × 0.05 = 0.8 cents
- For 4×5×5: Cost = (130/100) × 0.05 = 0.65 cents
From the calculations above, the least cost to make 100 boxes is 0.65 cents for the dimensions 4×5×5.