Answer:
[tex]Length =\frac{25}{4 + \pi}[/tex] and [tex]Width = \frac{50}{4+\pi}[/tex]
Step-by-step explanation:
This question is better understood with an attachment.
See attachment for illustration.
Given
Represent Perimeter with P
[tex]P = 25ft[/tex]
Required
Determine the dimension of the rectangle that maximizes the area
First, we calculate the perimeter of the rectangular part of the window.
From the attachment, the rectangle is not closed at the top.
So, The perimeter would be the sum of the three closed sides
Where
[tex]Width = 2x[/tex]
[tex]Length = y[/tex]
So:
[tex]P_{Rectangle} = y + y + 2x[/tex]
[tex]P_{Rectangle} = 2y + 2x[/tex]
Next, we determine the circumference of the semi circle.
Circumference of a semicircle is calculated as:
[tex]C = \frac{1}{2}\pi r[/tex]
From the attachment,
[tex]Radius (r) = x[/tex]
So, we have:
[tex]C = \frac{1}{2}2\pi * x[/tex]
[tex]C = \pi x[/tex]
So, the perimeter of the window is:
[tex]P = P_{Rectangle} + C[/tex]
[tex]P =2y + 2x + \pi x[/tex]
Recall that: [tex]P = 25[/tex]
So, we have:
[tex]25 =2y + 2x +\pi x[/tex]
Make 2y the subject
[tex]2y = 25 - 2x - \pi x[/tex]
Make y the subject:
[tex]y = \frac{25}{2} - \frac{2x}{2} - \frac{\pi x}{2}[/tex]
[tex]y = \frac{25}{2} - x - \frac{\pi x}{2}[/tex]
Next, we determine the area (A) of the window
A = Area of Rectangle + Area of Semicircle
[tex]A = 2x * y + \frac{1}{2}\pi r^2[/tex]
[tex]A = 2xy + \frac{1}{2}\pi r^2[/tex]
Recall that
[tex]Radius (r) = x[/tex]
[tex]A = 2xy + \frac{1}{2}\pi x^2[/tex]
Substitute [tex]\frac{25}{2} - x - \frac{\pi x}{2}[/tex] for y in [tex]A = 2xy + \frac{1}{2}\pi x^2[/tex]
[tex]A = 2x(\frac{25}{2} - x - \frac{\pi x}{2}) + \frac{1}{2}\pi x^2[/tex]
Open Bracket
[tex]A = 2x * \frac{25}{2} - 2x * x - 2x * \frac{\pi x}{2} + \frac{1}{2}\pi x^2[/tex]
[tex]A = 25x - 2x^2 - \pi x^2 + \frac{1}{2}\pi x^2[/tex]
[tex]A = 25x - 2x^2 - \frac{1}{2}\pi x^2[/tex]
To maximize area, we have to determine differentiate both sides and set A' = 0
Differentiate
[tex]A' = 25 - 4x - \pi x[/tex]
[tex]A' = 0[/tex]
So, we have:
[tex]0 = 25 - 4x - \pi x[/tex]
Factorize:
[tex]0 = 25 -x(4 + \pi)[/tex]
[tex]-25 =-x(4 + \pi)[/tex]
Solve for x
[tex]x = \frac{-25}{-(4+\pi)}[/tex]
[tex]x = \frac{25}{4+\pi}[/tex]
Recall that
[tex]Width = 2x[/tex]
[tex]Width = 2(\frac{25}{4+\pi})[/tex]
[tex]Width = \frac{50}{4+\pi}[/tex]
Recall that:
[tex]y = \frac{25}{2} - x - \frac{\pi x}{2}[/tex]
Substitute [tex]\frac{25}{4+\pi}[/tex] for x
[tex]y = \frac{25}{2} - (\frac{25}{4+\pi}) - \frac{\pi (\frac{25}{4+\pi})}{2}[/tex]
[tex]y = \frac{25}{2} - (\frac{25}{4+\pi}) - \frac{\frac{25\pi}{4+\pi}}{2}[/tex]
[tex]y = \frac{25}{2} - (\frac{25}{4+\pi}) - \frac{25\pi}{4+\pi} * \frac{1}{2}[/tex]
[tex]y = \frac{25}{2} - \frac{25}{4+\pi} - \frac{25\pi}{2(4+\pi)}[/tex]
[tex]y = \frac{25(4+\pi) - 25 * 2 - 25\pi}{2(4 + \pi)}[/tex]
[tex]y = \frac{100+25\pi - 50 - 25\pi}{2(4 + \pi)}[/tex]
[tex]y = \frac{100- 50+25\pi - 25\pi}{2(4 + \pi)}[/tex]
[tex]y = \frac{50}{2(4 + \pi)}[/tex]
[tex]y = \frac{25}{4 + \pi}[/tex]
Recall that:
[tex]Length = y[/tex]
So:
[tex]Length =\frac{25}{4 + \pi}[/tex]
Hence, the dimension of the rectangle is:
[tex]Length =\frac{25}{4 + \pi}[/tex] and [tex]Width = \frac{50}{4+\pi}[/tex]
