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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