a rectangular box with a square bottom is to have a volume of 1000 cubic inches. the bottom cost $100 per cubic inch, the sides are $25 per square inch and there is no top. what are the dimensions of the box that has materials costing the least

I bet the bottom is 100/in^2, not cubic.

size of box: s^2 * h s is side dimention on bottom.

cost= 100*s^2+ 4*25sh

but h= 1000/s^2

cost= 100s^2+100s(1000/s^2)

dcost/ds= 200s-100*1000/s^2=0

200s^3=100,000
s= cuberoot(500)=7.94in
h= 1000/7.94^2=15.9in

To find the dimensions of the box that has the least cost, we need to determine the relationship between the dimensions and the cost. Let's break it down step by step:

1. Let's assume the dimensions of the square bottom are both "x" inches. Therefore, the area of the square bottom is x * x = x^2 square inches.

2. The volume of the rectangular box is given as 1000 cubic inches. Since the bottom of the box is square, the height of the box would also be "x" inches.

3. The cost of the bottom is $100 per cubic inch, and the volume of the bottom is x^2 * x = x^3 cubic inches. So, the cost of the bottom would be 100 * x^3 dollars.

4. The remaining four sides of the box (excluding the bottom, since there is no top) form a rectangle. The total surface area of the four sides is 2 * (length * width) = 2 * (x * x) = 2x^2 square inches.

5. The cost of the four sides is $25 per square inch. So, the cost of the sides would be 25 * (2x^2) = 50x^2 dollars.

6. Since we want to minimize the cost, we need to find the dimensions that minimize the total cost, which is the sum of the cost of the bottom and the cost of the sides:

Total Cost = Cost of Bottom + Cost of Sides
= 100 * x^3 + 50x^2

7. To find the dimensions that minimize the cost, we need to find the value of "x" that minimizes the total cost. We can do this by taking the derivative of the total cost with respect to "x" and setting it equal to zero:

d(Total Cost)/dx = 0

Differentiating the total cost equation:
d(Total Cost)/dx = 300x^2 + 100x - 100 = 0

Solving the above equation gives us the value of "x".

Once we have the value of "x", we can calculate the width and length of the box by using x as all sides of the square bottom and the height of the box.

By applying these steps, we can find the dimensions of the box that has the least cost.