Question

For a window consisting of a rectangular pane surmounted by a semicircular pane, if the perimeter is 20 feet, what are the dimensions that maximize the area?

a) Equal dimensions for both panes

b) Rectangular pane height equals semicircular pane radius

c) Rectangular pane width equals semicircular pane diameter

d) Rectangular pane width equals semicircular pane radius

Answer 1

To maximize the area of the window, we can use calculus to find the critical points of the quadratic equation. By plugging these points into the equation, we can find the dimensions that maximize the area.

Step-by-step explanation:

To find the dimensions that maximize the area of the window, we can use calculus. Let's assume the rectangular pane has width x and height h, and the semi circular pane has radius r. The perimeter is given as 20 feet, so we can set up the equation 2x + h + πr = 20. We need to express the area, A, in terms of one variable, so we can substitute h = 20 - 2x - πr into the equation for the area, A = xh + (1/2)πr².

Now, we have A = x(20 - 2x - πr) + (1/2)πr². Simplifying this equation will give us a quadratic equation in terms of x, which we can then maximize using calculus. By taking the derivative of A with respect to x and setting it equal to 0, we can find the critical points. By plugging the critical points into the equation for A, we can find the dimensions that maximize the area of the window.