2. A farmer has a water trough that they would like to expand. The picture attached shows a cross-section of the configuration of their current water trough and plans for a future water trough.
(a) Write the area, A(θ), as a function of the variable θ using sinθ and cosθ. Note: The notation A(θ) indicates that θ should be the only variable that appears in your function for the area. It will also be helpful to simplify your function as much as possible.
(b) Fill in the table below to describe the area for various different angles of θ.
θ 0◦ 5◦ 15◦ 25◦ 35◦ 45◦ 55◦ 65◦ 75◦ 85◦
A(θ)
(c) Draw a graph of A(θ) on a reasonable domain that makes sense for this situation. (d) Explain the meaning of the y-intercept of the graph.
(e) Use the maximum function on a graphing calculator to fill in the blanks: The maximum area of a cross-section of the water trough is (BLANK) and the angle that gives that maximum is (BLANK).
