SOLUTION: What is the radius if: degree of measurement of arc=30 length=1/3xy(pi) My answer was 8xy. degree of measurement of arc=40 length=8/9(t)(pi) My answer was 9t. degree

Question 314082: What is the radius if:
degree of measurement of arc=30
length=1/3xy(pi)
My answer was 8xy.
degree of measurement of arc=40
length=8/9(t)(pi)
My answer was 9t.
degree of measurement of arc=18
length=6(y)pi
My answer was 45y.

These are the questions I missed, along with my answers. I would be grateful if someone could explain what I did wrong and work out the problem, so that I may understand why I missed these questions. Cheers!

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
The circumference of a circle is equal to 2*pi*r

The measurement of the arc of a circle is equal to the degree of the arc divided by 360 times the circumference of the circle.

Since the circumference of the circle is equal to 2 * pi * r, this means that the measurement of the are of the circle is equal to the degree of the arc divided by 360 times 2 * pi * r.

If you let L = the length of the arc and D equal the degree of the arc, then the equations you should be working with are:

L = D/360 * C which is equivalent to L = D/360 * 2 * pi * r

Then all you have to do is solve for r.

In your first problem, you are given:
What is the radius if:
degree of measurement of arc=30
length=1/3xy(pi)

The equation you have to work with is L = D/360 * 2 * pi * r

Substituting in this equation, you get:

1/3 * x * y * pi = 30/360 * 2 * pi * r

Simplify this equation to get:

1/3 * x * y * pi = 1/6 * pi * r

Divide both sides of this equation by pi to get:

1/3 * x * y = 1/6 * r

Multiply both sides of this equation by 6 to get:

2 * x * y = r

That's your answer.

To confirm, substitute for r in the equation for the circumference of the circle to get:

C = 2 * pi * r = 2 * pi * 2 * x * y = 4 * pi * x * y

Since L = D/360 * C, then substitute for C to get:

L = D/360 * 4 * pi * x * y

Substitute for D to get:

L = 30/360 * 4 * pi * x * y

Simplify to get:

L = 4/12 * pi * x * y which simplifies further to L = 1/3 * pi * x * y

Since this is the length of the arc you started with, then you are good.

In your second problem, you are given:
degree of measurement of arc=40
length=8/9(t)(pi)

The equations you need to work with are L = D/360 * C and L = D/360 * 2 * pi * r.

To find the radius, use L = D/360 * 2 * pi * r

Substitute for L to get:

8/9 * t * pi = 40/360 * 2 * pi * r

Simplify to getr:

8/9 * t * pi = 2/9 * pi * r

Divide both sides of this equation by pi to get:

8/9 * t = 2/9 * r

Multiply both sides of this equation by 9/2 to get:

4*t = r

That should be your answer.

If r = 4*t then C = 2 * pi * r = 2 * pi * 4 * t = 8 * pi * t

Since L = D/360 * C, substitute 40 for D and 8 * pi * t for C to get:

L = 40/360 * 8 * t * pi which simplifies to 8/9 * t * pi.

Since this is the length of the arc you started with, then you are good.

In your third problem, you are given:

degree of measurement of arc=18
length=6(y)pi

Use the formula L = D/360 * 2 * pi * r to get:

6 * y * pi = 18/360 * 2 * pi * r

Simplify to get:

6 * y * pi = 1/20 * 2 * pi * r which simplifies further to get:

6 * y * pi = 1/10 * pi * r

Divide both sides of this equation by pi to get:

6 * y = 1/10 * r

Multiply both sides of this equation by 10 to get:

60 * y = r

That's your answer.

Since C = 2 * pi * r, then you get:

C = 2 * pi * 60 * y which simplfies to C = 120 * pi * y

Since L = D/360 * C, then you get:

L = 18/360 * 120 * pi * y which simplifies to L = 120/20 * pi * y which simplifies further to:

L = 6 * pi * y.

Since this is the length of the arc you started with, they you are good.

The key to solving this problem is to use the equations:

C = 2 * pi * r

L = D/360 * C

C is equal to the circumference of the circle.
L is equal to the length of the arc
r is equal to the radius of the circle.
D is equal to the degree of the arc.

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