Question

An exercise room consists of a rectangle with a semicircle on each end. A 200-meter running track runs around the outside of the room. (a) Draw a figure to represent the problem. Let $x$ and $y$ represent the length and width of the rectangle. (b) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) Use the table to guess the maximum area of the rectangular region. (c) Write the area $A$ as a function of $x$ . (d) Use calculus to find the critical number of the function in part (c) and find the maximum value. (e) Use a graphing utility to graph the function in part (c) and verify the maximum area from the graph.

          An exercise room consists of a rectangle with a semicircle on each end. A 200-meter running track runs around the outside of the room.
(a) Draw a figure to represent the problem. Let $x$ and $y$ represent the length and width of the rectangle.
(b) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) Use the table to guess the maximum area of the rectangular region.
(c) Write the area $A$ as a function of $x$ .
(d) Use calculus to find the critical number of the function in part (c) and find the maximum value.
(e) Use a graphing utility to graph the function in part (c) and verify the maximum area from the graph.

Added by Sean R.

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