Final answer:
To determine the dimensions of the box with the maximum volume, you need to find the dimensions of the square cut out from the corners and the volume function. By calculating the derivative of the volume function and setting it to zero, you can find the value of 'x' that maximizes the volume. Finally, substitute this value in the expressions for length, width, and height to obtain the dimensions of the box.
Step-by-step explanation:
Determining the Dimensions of the Box
To determine the dimensions of the box with the maximum volume, we first need to find the dimensions of the square cut out from the corners. Let's assume the side length of this square is 'x' inches. By cutting out squares from the corners, the length of the box will be reduced by 2x inches and the width will be reduced by 2x inches. Therefore, the length of the box will be (15 - 2x) inches and the width will be (18 - 2x) inches.
Expression for the Volume
The volume of the box can be found by multiplying the length, width, and height. In this case, the height of the box will be 'x' inches. Therefore, the volume V(x) of the box is given by:
V(x) = (15 - 2x)(18 - 2x)(x)
Finding the Maximum Volume
We can find the maximum volume by finding the value of 'x' that maximizes the volume function V(x). This can be done by calculating the derivative of V(x) with respect to 'x' and setting it equal to zero. After solving the resulting equation, we can substitute the value of 'x' into the expressions for the length, width, and height to find the dimensions of the box with the maximum volume.