1. A wire of length x is bent into the shape of a square. Express the area A as a function of x only.

2. A circle of radius x is inscribed in a square. Find the area inside the square but outside the circle as a function of x only.

1. the wire would have to be cut into 4 equal parts, making each side of the square x/4 units long

so Area = (x/4)^2 = x^2 /16

2. In this one, isn't the side of the square equal to 2x (the diameter of the circle) ?
You should be able to take it from here.

So the area would be 4x^2 ?

and the area of a circle is pi * x^2

So would the area be (4-pi) x^2?

yes

1. To express the area A as a function of x only, let's start by considering the shape of the square formed by the wire.

A square has four equal sides, so the length of each side of the square is x/4 (since the wire's length is x). The area of a square is given by the formula A = side length * side length.

Therefore, in this case, the area of the square A can be expressed as:

A = (x/4) * (x/4) = x^2/16

So, the area A of the square can be expressed as a function of x only, A(x) = x^2/16.

2. To find the area inside the square but outside the inscribed circle, we need to determine the areas of both the square and the circle, and then subtract the area of the circle from the area of the square.

The area of a square is given by the formula A = side length * side length. Since the side length of the square is equal to 2 times the radius of the circle (x), the area of the square is:

A_square = (2x) * (2x) = 4x^2

The area of a circle is given by the formula A = π * radius^2. In this case, the radius of the circle is x, so the area of the circle is:

A_circle = π * x^2

Now, to find the area inside the square but outside the circle, we subtract the area of the circle from the area of the square:

A_inside = A_square - A_circle
= 4x^2 - πx^2

Therefore, the area inside the square but outside the circle can be expressed as a function of x only, A_inside(x) = 4x^2 - πx^2.