The dimensions of the box could be . Then, the box has four sides with dimension and one side with dimension . Hence, . You want to maximize the volume, which is . Now, from the first equation, you should solve for : . Plugging that in to the volume formula:

Now, to maximize volume, take the derivative and set it equal to zero:

. So, . Check the second derivative: . Since , there is a local maximum at . Hence, the maximum area is given by:

so should be the correct answer.

First, I would recommend rounding at the very end. Second, the width of the window is , as you suggested. But, the height of the rectangular portion could be anything. So, let the height of the window be .

Then, area is:

Plugging in for , you have:

Then . Setting it equal to zero, you find . The second derivative is always negative, so the area of the window is maximized at that value. So, the maximum area is , so 148 square feet should be the correct answer.