You have a fixed amount of window trim for a particular window. The window is a rectangle with a circle removed, and the circle has a diameter equal to the width of the rectangle: Let P represent the fixed perimeter. Let r represent the radius of the circle. Let y represent the height of the rectangle. Let A represent the Area of the SHADED REGION, that is, the area inside the rectangle but OUTSIDE the circle. (a) Write a constraint involving the perimeter; that is, write P in terms of r and y. (b) Write an equation for the Area as a function of r and y. (c) Solve the constraint for y (or even better, 2y) and substitute into the Area equation.
You have a fixed amount of window trim for a particular window. The window is a rectangle with a circle removed, and the circle has a diameter equal to the width of the rectangle: Let P represent the fixed perimeter. Let r represent the radius of the circle. Let y represent the height of the rectangle. Let A represent the Area of the SHADED REGION, that is, the area inside the rectangle but OUTSIDE the circle. (a) Write a constraint involving the perimeter; that is, write P in terms of r and y. (b) Write an equation for the Area as a function of r and y. (c) Solve the constraint for y (or even better, 2y) and substitute into the Area equation.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:You have a fixed amount of window trim for a particular window. The window is a rectangle with a
circle removed, and the circle has a diameter equal to the width of the rectangle:
Let P represent the fixed perimeter.
Let r represent the radius of the circle.
Let y represent the height of the rectangle.
Let A represent the Area of the SHADED REGION, that is, the area
inside the rectangle but OUTSIDE the circle.
(a) Write a constraint involving the perimeter; that is, write P in terms of r and y.
(b) Write an equation for the Area as a function of r and y.
(c) Solve the constraint for y (or even better, 2y) and substitute into the Area equation.
(d) Take the derivative and set it equal to zero. Solve for P.
(e) Substitute for P in the equation in part (c) and solve for y.
(f) What is the RATIO of the height y to width 2r for the rectangle that maximizes the shaded area?
(g) Interpret the geometric meaning of the result in (f).
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