# A company wants to bid on an order of 20,000 square based boxes which hold 5 cubic feet. The cardboard cost is $.15 per square foot. The boxes must be reinforced with an extra layer of cheaper cardboard on the bottom ($.12 per square foot). The box is held together with tape on all four edges and one side edge. There is also one edge along the top which is taped to create a hinge. It cost $.10 per linear foot to tape edges.

1. Write a function for the cost of a box based on the length of the base.

2. Graph your cost function.

3. What should the dimensions of the box be in order to minimize the cost?

4. How much will it cost to make the 20,000 boxes?

5. If the company wants to make 15% what should they bid?

13 years ago

13 years ago

## I am having difficulty coming up with the right function. The rest should fall into place, I know.

13 years ago

## area = 2 ends + 2 sides +extra bottom+ bottom+ edge+top this assumes top,two bottoms, and four sides. I assume the four edges mentioned are the three bottom edges, and one side edge.

cost=area*cost/srea= 2*h*w*.15+ 2*h*l*.15+ l*w*.15+l*w*.12+ w*l*.15 +.10(2w+h+l+w)

simplify that into a function of l,w,h

Then volume=5=lwh and you can solve h=5/lw

check carefully the area and cost function. A sketch of the box might help. I assume the bottom was hinged on one side, taped on three sides, then tape on one height side, and finally, tape for a folding lid. Weird box.

13 years ago

## remember it is square, so w=l

13 years ago

## That is what I came up with for my function. It just didn't seem to work out right when I went to solve. I will have to go back and check my math. Thank you for your help!!!

10 months ago

## 1. To write a function for the cost of a box based on the length of the base, we need to consider the different cost factors involved. Let's break it down step by step:

- The cost of the cardboard for the sides of the box (excluding the bottom) is given by the area of the sides (2 * length * height + 2 * width * height) multiplied by the cost per square foot.

- The cost of the bottom reinforcement cardboard is given by the area of the base (length * width) multiplied by the cost per square foot.

- The cost of the tape is given by the perimeter of the base (2 * (length + width)) plus the length of the top edge, multiplied by the cost per linear foot.

- To get the total cost, we sum up the above three costs.

Therefore, the function for the cost of a box based on the length of the base (let's call it "C") can be written as:

C(length, width, height) = (2 * length * height + 2 * width * height + length * width) * $.15 + (length * width) * $.12 + (2 * (length + width) + length) * $.10

2. To graph the cost function, we can fix the height at a constant value and represent the cost as a function of the base dimensions (length and width). We can create a 2-dimensional graph with the length on the x-axis and the cost on the y-axis. We can vary the length and width within certain ranges to observe the cost variations.

3. To minimize the cost, we need to find the dimensions of the box that result in the lowest total cost. This can be achieved by using mathematical optimization techniques. In this case, we can take the partial derivatives of the cost function with respect to the length and width, set them equal to zero, and solve for the values of length and width that minimize the cost.

4. To determine the cost of making 20,000 boxes, we would need to multiply the cost of one box (computed using the function from step 1) by the number of boxes (20,000). This will give us the total cost of making all the boxes.

5. To calculate the bid amount, the company needs to consider the cost of making the boxes and their desired profit margin. The bid amount should be the total cost of making the boxes plus the desired profit margin (which is a percentage of the cost). The specific percentage to use for profit margin can vary based on factors such as market conditions and company policies.