please help me with this question

Question

A wire of length

8x

is bent into the shape of a circle.

A. Express the circumference of the circle as a function of x.

B. Express the area of the circle as a function of x.

Answer 1

Hi Mary C.,

For A., the circumference of a circle is equal to the diameter times pi or the radius times 2 pi.

c = d*π or c = 2*π*r. You can intuitively deduce that the length of the wire is equal to the circumference (c = 8x) or algebraically you have:

c = 2*π*r

8x = 2*π*r, (not a function of x)

8x/2π = r, (solve for r, divide both sides by 2π, we need this for step B)

c = 2*π*(8x/2π), (plug r back into c = 2*π*r)

c = 8x, (the circumference is a function of x)

For B, the area of a circle is equal to the radius squared time pi, A=π*r².

A = π*(8x/2π)², (r = 8x/2π from A.)

A = π*(64x²/4π²), (all squared)

A = 64x²/4π, (π's cancel)

A = 16x²/π, (the area as a function of x)

I hope this helps, Joe.

Answer 2

Since the wire is bend into the shape of the circle, circumference of circle is same as the length.

Circumference=8x.

We also know that, circumference =2*πr

So, 2πr=8x

r=4x/π

Area=πr²= π(4x/π)²

Area=16x²/π

Answer 3

The circumference is the DISTANCE around the circle. The CIRCUMFERENCE of a circle is pi*diameter. So we have C = pi*d, which is 8x. So we have the DIAMETER = 8*x/pi. A RADIUS is 1/2 of a diameter. In this case, the RADIUS = 4x/pi. The AREA of a CIRCLE is pi*r^2. In this case the AREA is pi*(4x/pi)^2. This simplifies to (16*pi*x^2) / pi^2 or 16*x^2 OVER pi^2. The final result is 16*x^2 OVER pi. Reiterating the CIRCUMFERENCE of this CIRCLE is 8*x; the DIAMETER is 8*x/pi; the RADIUS is 4*x/pi; and the AREA of this circle is 16*x^2/ pi.

Answer 4