a semicircle of radius r=3x is inscribed in a rectangle so that the diameter of the semicircle is the length of the rectangle

1. express the area A if the rectangke as a function
2 express the perimeter P of the rectamgle as a function of x

1. Area A of the rectangle can be expressed as A(x) = (2x)(4x) + (πr^2)/2, where r = 3x. Simplifying this equation, we get A(x) = 8x^2 + (9πx^2)/2.

2. Perimeter P of the rectangle can be expressed as P(x) = 2(2x) + 2(4x) + πr, where r = 3x. Simplifying this equation, we get P(x) = 8x + 16x + 3πx.

1. To express the area, A, of the rectangle as a function of x, we need to find the length and width of the rectangle.

The diameter of the semicircle is the length of the rectangle, so the length is 2(3x) = 6x.

Since the rectangle is inscribed in the semicircle, the width of the rectangle is equal to the radius of the semicircle, which is 3x.

Therefore, the area A of the rectangle is given by the formula: A = length * width.
Substituting the values obtained, we have: A = 6x * 3x = 18x^2.

So, the area A of the rectangle is a function of x and can be expressed as A(x) = 18x^2.

2. To express the perimeter, P, of the rectangle as a function of x, we add up all the sides of the rectangle.
The length of the rectangle is 2(3x) = 6x, and the width is 3x.

The perimeter P of the rectangle is given by the formula: P = 2(length + width).
Substituting the values obtained, we have: P = 2(6x + 3x) = 2(9x) = 18x.

So, the perimeter P of the rectangle is a function of x and can be expressed as P(x) = 18x.

To summarize:
1. Area A of the rectangle as a function of x is A(x) = 18x^2.
2. Perimeter P of the rectangle as a function of x is P(x) = 18x.

To solve this problem, let's go step by step:

1. Express the area A of the rectangle as a function:

The diameter of the semicircle is equal to the length of the rectangle, which means it has a length of 2r (since the radius is r). The width of the rectangle is equal to the radius of the semicircle, which is r. Therefore, the dimensions of the rectangle are 2r and r.

The formula for the area of a rectangle is A = length × width. In this case, it becomes A = 2r × r = 2r^2.

However, since r = 3x, we can substitute it back into the equation to get the area as a function of x:

A = 2(3x)^2 = 2(9x^2) = 18x^2

So, the area of the rectangle, A, is given by the function A(x) = 18x^2.

2. Express the perimeter P of the rectangle as a function of x:

The perimeter of a rectangle is calculated by adding the lengths of all four sides. In this case, the length is 2r and the width is r.

The formula for the perimeter of a rectangle is P = 2(length + width). Substituting the dimensions of the rectangle, we have:

P = 2(2r + r) = 2(3r)

Since r = 3x, we can substitute it back into the equation to get the perimeter as a function of x:

P = 2(3(3x)) = 2(9x) = 18x

So, the perimeter of the rectangle, P, is given by the function P(x) = 18x.

Clearly, since we have a semicircle, the width of the rectangle is half the length. So,

A = (3x/2)(3x)
P = 2(3x/2 + 3x)