Question

A circle's center is at `(3 ,1 )` and it passes through `(6 ,6 )`. What is the length of an arc covering `(7pi ) /12` radians on the circle?

Answer 1

`(7sqrt(34)/12)pi`

Explanation:

(WARNING: May be grossly worded, I will try to fix it later)

Okay so you have a circle. You want an arc length. Arc length is a portion of the circumference of a circle, so we need to find the circumference and then find the arc length (7`pi`/12 radians) from it.

We know two points of the circle. We know the center (3,1), and we know a point the circle passes through (6,6). Since we know a point on the circle and the center, we can use them to find the radius, since a radius is any line on a circle that goes from the center to any point on the circle.

So we need to use the distance formula:

`sqrt((X_1-X_2)^2+(Y_1-Y_2)^2)`

Plug in the coordinates, where (6,6) is our `X_1` and `Y_1` and (3,1) is our `X_2` and `Y_2`:

`sqrt((6-3)^2+(6-1)^2) = sqrt(34)`

Now we have the radius. Our goal is still to find the circumference. To get the circumference with the radius, we use the formula for circumference:

2`pi`r

Plug in radius:

`2pi(sqrt(34)) = 2sqrt(34)pi`

Okay, now we reached our first goal, which was to find the circumference. Now we can move on to finding the arc length.

Arc length is a fraction of the circumference. When they gave us the radians for the arc length, they gave us the fraction of the total radians for the circumference. A circle's total arc length in radians is 2`pi`. So we need to find the fraction that represents 7`pi`/12 of 2`pi`, aka what fraction of the total arc length is 7`pi`/12:

The fraction 7/12 from 7`pi`/12 is talking about 7 parts out of the 12 parts of a semicircle (because a semicircle's arc length is `pi`, and the arc length is 7/12 of `pi` [shown as 7`pi`/12]), which means that we need to think about it in terms of the whole circle . If the semicircle has 12 parts, the whole circle will have 24 parts, so we are really looking at 7/24 as our fraction. 7 parts out of 24. Now we have a fraction that takes into account all the parts of a circle.

We now have a fraction to use on our circumference. We want to know the arc length that takes up 7 parts of 24, so we multiply 7/24 by the circumference, the circumference being the total arc length.

`7/24 * 2sqrt(34)pi`

If you simplify that, you will get the answer:

`(7sqrt(34)/12)pi`

Answer 2

When the central angle `theta` is defined in radians, the arc length:
`L = theta*r`.

For this problem:
`L=(7pi)/12*sqrt34`

Explanation:

The circumference of a complete circle is `2pir` and the total angle, in radians, is `2pi`.

The length of an arc is a portion of the complete circumference, given by the ratio of the central angle `theta` over the total angle of the circle.

So, the arc length: `L = theta/(2pi)*2pir`
When we are working with radians, this simplifies to:
`L=theta*r`

If we know the central angle, then we only need to find the radius of the circle, which is the distance from the center to any point on the circle.

In this problem, the central angle `theta=(7pi)/12` and the radius of the circle is the distance between the 2 points given: `(3,1)` and `(6,6)`

We can use Pythagoras theorem to obtain this distance:
`r^2 = (Delta x)^2 + (Delta y)^2`
`r^2 = (6-3)^2 + (6-1)^2`
`r^2 = 9 + 25 = 34`
`r = sqrt34`

So: `L=(7pi)/12*sqrt34`