Answer:
and
Step-by-step explanation:
This question is better understood with an attachment.
See attachment for illustration.
Given
<em>Represent Perimeter with P</em>
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
So:
Next, we determine the circumference of the semi circle.
Circumference of a semicircle is calculated as:
From the attachment,
So, we have:
So, the perimeter of the window is:
Recall that:
So, we have:
Make 2y the subject
Make y the subject:
Next, we determine the area (A) of the window
A = Area of Rectangle + Area of Semicircle
Recall that
Substitute for y in
Open Bracket
To maximize area, we have to determine differentiate both sides and set A' = 0
Differentiate
So, we have:
Factorize:
Solve for x
Recall that
Recall that:
Substitute for x
Recall that:
So:
Hence, the dimension of the rectangle is:
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