Let x be the width, which is equal to the diameter of the semi-circle. Then the perimeter around just the semi-circle is (1/2) pi x. Let y the height of the rectangular portion of the window. Perimeter around just the rectangular portion of the window is x + 2y. The total perimeter is (1/2) pi x + x + 2y = 30 Solve this equation for y: 2y = 30 - (1/2) pi x - x y = 15 - (1/4) pi x - x/2 Then the area of the rectangular portion is xy. The area of the semi-circle is (1/2) pi (x/2)^2. The total area = A = (1/2) pi (x/2)^2 + xy Substitute the expression for y found above into this last equation: A = (1/2) pi (x/2)^2 + x(15 - (1/4) pi x - x/2 ) Simplify and combine like terms: A = x^2(-pi - 4)/8 + 15x Take the derivative and set it to zero: A' = (1/4) (-4-pi)x + 15 = 0 Solve for x: (1/4) (-4-pi)x = -15 Multiply by -4: (4+pi)x = 60 Divide: x = 60 / (4+pi) ≈ 8.4 ft y = 15 - (1/4) pi x - x/2 = 30 / (4+pi) ≈ 4.2 ft
The x represents the ticket prices and the y represents the popcorn price. If we distribute the expression, we get 4x +2x + 2y. This shows us that all 6 people bought a ticket, and two bought popcorn. Now plug in the given values and solve.
