Answer:
The maximum volume of the box is 252 cubic inches.
Explanation:
To find the maximum volume of the box, we use the formula for the volume of a rectangular prism.
We want to maximize the volume while keeping the height of the box equal to the height of the original cardboard, which is 9 inches.
Let x be the length of the side of the square that is cut from each corner. The length of the box will be 15 - 2x and the width will be 9 - 2x.
Substituting these values into the volume formula, we get:
V = (15 - 2x)(9 - 2x)x
Expanding the parentheses and simplifying, we get:
V = 135x - 72x^2 + 4x^3
To find the maximum volume, we take the derivative of V with respect to x and set it equal to 0:
dV/dx = 12x^2 - 144x + 12
Factoring the expression, we get:
dV/dx = 12(x - 6)(x - 2)
Setting dV/dx equal to 0, we get the critical points x = 6 and x = 2.
Since V is a polynomial, it is defined for all real numbers.
Therefore, our critical points are within the domain of the function.
We can use the second derivative test to determine whether these critical points correspond to maxima or minima:
d^2V/dx^2 = 24(x - 4)
Evaluating d^2V/dx^2 at x = 6, we get:
d^2V/dx^2 = 24(6 - 4) = 48
Since d^2V/dx^2 > 0 at x = 6, we know that V is concave up at this point, which means that V is minimized at x = 6.
This means that the maximum volume is achieved when the side of the square that is cut from each corner is 6 inches.
Plugging x = 6 into the volume formula, we get:
V = 135(6) - 72(6)^2 + 4(6)^3 = 252
Therefore, the maximum volume of the box is 252 cubic inches.