Question

I need the answers for all four questions on that sheet. Including shown work

Answer 1

1. 8.4 miles

2. "I traveled further, by 1.3 miles"

3. 10.2 square miles

4. 18.8 miles

Explanation:

1.

The path traveled is the distance along the circle from Tennis Court to Nature Center. This is part of the circumference, we call it "arc length". The formula for arc length is:

Arc Length =
`(\theta)/(360)*2\pi r`

Where

`\theta` is the angle traversed

and

r is the radius of the circle

From Tennis Court to Nature center, the angle is 120 + 40 = 160

and radius is given as 3

So we substitute and find the answer:

`(\theta)/(360)*2\pi r\\=(160)/(360)*2(3.14) (3)\\=8.4`

So he traveled 8.4 miles

2.

We follow similar process as #1.

From pool to nature center is 45 + 90 = 135 degrees

The radius is still 3

So we substitute into arc length formula and figure her length:

`(\theta)/(360)*2\pi r\\=(135)/(360)*2(3.14)(3)\\=7.1`

So, Friend traveled 7.1 miles

So, I traveled further, by 8.4 - 7.1 = 1.3 miles

3.

Here, we need Part of Area of the circle, that is basically "area of a sector of a circle". The formula for this is:

Sector Area =
`(\theta)/(360)*\pi r^2`

Where

`\theta` and r represents the same.

The angle subtended from Ranger Station to Nature Center is 90 + 40 = 130 degrees

The radius is 3

So, we put into formula and figure out the area of that part:

`(\theta)/(360)*\pi r^2\\=(130)/(360)*(3.14) (3)^2\\=10.2`

The area is 10.2 sq. mi.

4.

The total distance of the park (the perimeter) is basically the total length. This is known as the circumference. It has the formula shown below:

`C=2\pi r`

Where

C is the circumference

r is the radius

Given, radius is 3, we put into formula and find our answer:

`C=2\pi r\\C=2(3.14)(3)\\C=18.8`

That is, 18.8 miles