Answer:
Part 1) ![A=124.63\ in^{2}](https://tex.z-dn.net/?f=A%3D124.63%5C%20in%5E%7B2%7D)
Part 2) Option b
Part 3) As n increases, ns get closer to
Part 4) Option c ![16\pi\ ft^{2}](https://tex.z-dn.net/?f=16%5Cpi%5C%20ft%5E%7B2%7D)
Part 5) Option b. ![12.25\pi\ m^{2}](https://tex.z-dn.net/?f=12.25%5Cpi%5C%20m%5E%7B2%7D)
Step-by-step explanation:
Part 1) What is the area of a circle with a diameter of 12.6 in.?
we know that
the area of a circle is equal to
we have
-----> the radius is half the diameter
substitute the values
Part 2) Which explanation can be used to derive the formula for the circumference of a circle?
First find the relationship of the circumference to its diameter by finding that the length of the diameter wraps around the length of the circumference approximately π times.
Use this relationship to write an equation showing the ratio of circumference to diameter equaling π
so
![\frac{C}{D}=\pi](https://tex.z-dn.net/?f=%5Cfrac%7BC%7D%7BD%7D%3D%5Cpi)
Rearrange the equation to solve for the circumference
![C=\pi D](https://tex.z-dn.net/?f=C%3D%5Cpi%20D)
Substitute the diameter for 2 times the radius
![D=2r](https://tex.z-dn.net/?f=D%3D2r)
Part 3) we know that
If n increases
then
the product ns get closer to the circumference of the circle
so
the circumference of a circle is equal to
therefore
As n increases, ns get closer to
Part 4) What is the area of a circle whose radius is 4 ft?
we know that
the area of a circle is equal to
![A=\pi r^{2}](https://tex.z-dn.net/?f=A%3D%5Cpi%20r%5E%7B2%7D)
we have
substitute the values
![A=(\pi)(4^{2})=16\pi\ ft^{2}](https://tex.z-dn.net/?f=A%3D%28%5Cpi%29%284%5E%7B2%7D%29%3D16%5Cpi%5C%20ft%5E%7B2%7D)
Part 5) The circumference of a circle is 7π m.
What is the area of the circle?
we know that
The circumference of a circle is equal to
![C=2\pi r](https://tex.z-dn.net/?f=C%3D2%5Cpi%20r)
we have
substitute and solve for r
![7\pi=2\pi r](https://tex.z-dn.net/?f=7%5Cpi%3D2%5Cpi%20r)
![r=3.5\ m](https://tex.z-dn.net/?f=r%3D3.5%5C%20m)
Find the area of the circle
the area of a circle is equal to
![A=\pi r^{2}](https://tex.z-dn.net/?f=A%3D%5Cpi%20r%5E%7B2%7D)
substitute
the area of a circle is equal to
![A=\pi (3.5^{2})=12.25\pi\ m^{2}](https://tex.z-dn.net/?f=A%3D%5Cpi%20%283.5%5E%7B2%7D%29%3D12.25%5Cpi%5C%20m%5E%7B2%7D)