The Norman window illustrated below has a semicircular section on top of a rectangle. The radius of the semicircle is x. The long side of the rectangle is three times the radius of the semicircle.

a)Express the total area of the window, A, as a function of x.
b) Express the outer perimeter of the window, P, as a function of x.

area of window=x*3x+PI*x^2

Perimeter=2*3x+x+PI*r

This is hardly calculus.

The rectangle is 2x by 3x with the semicircle on top of the 2x side

Since there is a semicircle on top of the rectangle, the area

= rectangle + semicircle
= basexheight + (1/2)π x^2
= (2x)(3x) + π x^2/2
= 6x^2 + πx^2/2

perimeter = (1/2)(2πx) + 2x + 2(3x)
= πx + 8x

a) To find the total area, we need to calculate the area of the rectangle and add it to the area of the semicircle.

The area of the rectangle is given by length * width. Since the length of the rectangle is three times the radius of the semicircle, this can be expressed as 3x * x = 3x^2.

The area of the semicircle is given by 1/2 * π * radius^2. In this case, the radius is x, so the area of the semicircle becomes 1/2 * π * x^2.

Adding the area of the rectangle and the semicircle together, we get:

A = 3x^2 + 1/2 * π * x^2

Therefore, the total area of the window, A, can be expressed as a function of x as A(x) = 3x^2 + 1/2 * π * x^2.

b) The outer perimeter of the window is the sum of the lengths of all the sides. We can calculate this by adding the lengths of the three sides of the rectangle and the curved length of the semicircle.

The length of the three sides of the rectangle is given by 2(length + width), which in this case is 2(3x + x) = 2(4x) = 8x.

The curved length of the semicircle is half the circumference of a full circle with radius x. The circumference of a circle is given by 2πr, so the curved length of the semicircle becomes 1/2 * 2π * x = πx.

Adding the lengths of all the sides together, we get:

P = 8x + πx

Therefore, the outer perimeter of the window, P, can be expressed as a function of x as P(x) = 8x + πx.

To find the area and perimeter of the Norman window, let's break it down into its individual parts - the semicircle and the rectangle.

a) The area of the window, A, can be calculated by summing the areas of the semicircle and the rectangle. The formula for the area of a circle is A = πr^2, where r is the radius. However, since we have a semicircle, we need to multiply the result by 1/2.

The area of the semicircle is therefore A_semicircle = (1/2)πx^2.

The area of the rectangle is given by the formula A_rectangle = length * width. Since the width of the rectangle is the same as the diameter of the semicircle (2x), and the length of the rectangle is three times the radius (3x), we can write A_rectangle = 3x * 2x = 6x^2.

Thus, the total area of the window is A = A_semicircle + A_rectangle = (1/2)πx^2 + 6x^2.

b) The outer perimeter of the window, P, can be determined by summing the lengths of the semicircle and the rectangle.

The circumference of a semicircle is given by the formula C_semicircle = πd, where d is the diameter. In this case, the diameter is 2x, so C_semicircle = π(2x) = 2πx.

The perimeter of the rectangle is given by P_rectangle = 2(width + length). Since the width (diameter) of the rectangle is 2x and the length is 3x, we have P_rectangle = 2(2x + 3x) = 2(5x) = 10x.

Therefore, the outer perimeter of the window is P = C_semicircle + P_rectangle = 2πx + 10x.