# A circle has a chord that goes from #pi/3 # to #(3 pi) / 8 # radians on the circle. If the area of the circle is #81 pi #, what is the length of the chord?

By signing up, you agree to our Terms of Service and Privacy Policy

To find the length of the chord, we first need to determine the radius of the circle. Given that the area of the circle is (81\pi) and using the formula for the area of a circle ((A = \pi r^2)), we can solve for the radius ((r)).

[81\pi = \pi r^2]

Solving for (r):

[r^2 = 81]

[r = 9]

Now, we can find the length of the chord using the formula for the length of a chord in terms of the radius and the central angle:

[ \text{Chord Length} = 2r\sin\left(\frac{\theta}{2}\right) ]

Where ( r ) is the radius of the circle and ( \theta ) is the central angle subtended by the chord.

Given that the central angle ( \theta = \frac{3\pi}{8} - \frac{\pi}{3} = \frac{5\pi}{24} ), we can now calculate the length of the chord:

[ \text{Chord Length} = 2 \times 9 \times \sin\left(\frac{5\pi}{48}\right) ]

[ \text{Chord Length} = 18 \times \sin\left(\frac{5\pi}{48}\right) ]

[ \text{Chord Length} \approx 18 \times 0.2588 ]

[ \text{Chord Length} \approx 4.659 ]

Hence, the length of the chord is approximately (4.659) units.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- Two circles have the following equations #(x +5 )^2+(y +6 )^2= 36 # and #(x +2 )^2+(y -1 )^2= 81 #. Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?
- Two circles have the following equations: #(x +6 )^2+(y -5 )^2= 49 # and #(x -9 )^2+(y +4 )^2= 81 #. Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?
- A circle has a center that falls on the line #y = 5/2x +1 # and passes through #(1 ,2 )# and #(6 ,1 )#. What is the equation of the circle?
- A circle has a center at #(7 ,5 )# and passes through #(4 ,3 )#. What is the length of an arc covering #pi /4 # radians on the circle?
- A circle has a center that falls on the line #y = 8/7x +2 # and passes through # ( 2 ,1 )# and #(3 ,9 )#. What is the equation of the circle?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7