Final answer:
To maximize the area of the window, we need to find the dimensions of the rectangle and the semicircle. We can set up an equation using the given total feet of wood trim to solve for the dimensions of the rectangle and the semicircle. With the dimensions determined, we can then calculate the area of the window and find the maximum area by taking the derivative of the area formula.
Step-by-step explanation:
To maximize the area of the window, we need to find the dimensions of the rectangle and the semicircle.
Let's assume the width of the rectangle is x and the length is y. The perimeter of the rectangle would then be 2(x+y).
The semicircle is placed along the top of the rectangle, so its diameter is equal to the width of the rectangle, which is x.
The perimeter of the semicircle would be πx/2.
Since we are given the total feet of wood trim available for both the rectangle and the semicircle, we can set up an equation: 2(x+y) + πx/2 = total feet.
We can use this equation to express y in terms of x, substitute it back into the equation to get a quadratic equation in terms of x, and then solve for x to find the width of the rectangle and the diameter of the semicircle.
Once we have the dimensions, we can calculate the area of the window, which is the sum of the area of the rectangle and the area of the semicircle.
To find the maximum area, take the derivative of the area with respect to x, set it equal to zero, and solve for x. Then substitute this value back into the area formula to get the maximum area.