A Norman window is a rectangle with a semicircle on top. Suppose that the perimeter of a particular Norman window is to be 25 feet. What should the rectangle's dimensions be in order to maximize the area of the window and, therefore, allow in as much light as possible? (Round your answers to two decimal places.)width ftheight ft

and

Step-by-step explanation:

This question is better understood with an attachment.

See attachment for illustration.

Given

Represent Perimeter with P

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:

and

step-by-step explanation:

because 1/3 is a fraction. we can tell because there is a divisor symbol between the 1 and the 3. if we were to add 2 fractions together, we would add the numerators, which is the number on top, but not the denominators, which is the number on the bottom

1/3+1/3=2/3

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