an open top box is to be made by cutting congruent squares of side length x from the corners of a 12 by 15 inch sheet of tin and bending up the sides. how large should the squares be? what is the resulting maximum value?

To find the required size of the squares and the resulting maximum volume of the open top box, we need to follow these steps:

Step 1: Visualize the given information
We have a rectangular sheet of tin measuring 12 inches by 15 inches. We will cut squares of side length x from each corner of the sheet and fold up the sides to create an open-top box.

Step 2: Determine the dimensions of the box
By cutting squares of side length x from each corner, the dimensions of the resulting base will be 12 - 2x inches by 15 - 2x inches.

Step 3: Find the volume of the box
The volume of a box is calculated by multiplying the length, width, and height. In this case, the height of the box will be x inches.

Volume = length × width × height
= (12 - 2x) × (15 - 2x) × x

Step 4: Determine the maximum volume
To find the maximum volume, we need to find the value of x that maximizes the volume equation from Step 3.

Step 5: Algebraic solution for maximum volume
Differentiate the volume equation from Step 3 with respect to x to find its critical points:

dV/dx = 4x^3 - 54x^2 + 180x

Set the derivative equal to zero to find the critical points:

4x^3 - 54x^2 + 180x = 0

Factor out common terms:

4x(x^2 - 13.5x + 45) = 0

Apply the quadratic formula to solve the quadratic equation inside the parentheses:

x = (-b ± √(b^2 - 4ac)) / (2a)

Using a = 1, b = -13.5, and c = 45, we find:

x = (13.5 ± √(13.5^2 - 4(1)(45))) / (2(1))
x = (13.5 ± √(182.25 - 180)) / 2
x = (13.5 ± √(2.25)) / 2
x = (13.5 ± 1.5) / 2

This gives two possible solutions for x:

x1 = (13.5 + 1.5) / 2 = 7.5
x2 = (13.5 - 1.5) / 2 = 6

However, since the original problem states that the corners should be cut with congruent squares, x must be an integer value. Thus, x = 6 inches.

Step 6: Calculate the maximum volume
Now that we have found the value of x, substitute it into the volume equation from Step 3 to find the maximum volume:

Volume = (12 - 2x) × (15 - 2x) × x
= (12 - 2(6)) × (15 - 2(6)) × 6
= 6 × 3 × 6
= 108 cubic inches

Therefore, the required size of the squares to maximize the volume is 6 inches, and the resulting maximum volume of the open-top box is 108 cubic inches.