A circle has a center at #(7 ,9 )# and passes through #(4 ,3 )#. What is the length of an arc covering #pi /4 # radians on the circle?

Answer 1

I found #5.3#

We can use the 2 points to find the radius of the circle (evaluating the sdistance between them):

#r="distance between points"=sqrt((7-4)^2+(9-3)^2)=sqrt(45)=6.7#

The arc length #s# will be given as:
#s=rtheta=6.7*(3.14)/4=5.26~~5.3#

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Answer 2

To find the length of an arc covering ( \frac{\pi}{4} ) radians on the circle, you need to follow these steps:

  1. Calculate the radius of the circle using the distance formula between the center and the given point on the circle.
  2. Use the formula for the circumference of a circle to find the total circumference.
  3. Use the formula for arc length to find the length of the arc for ( \frac{\pi}{4} ) radians.

Let's proceed with the steps:

  1. Calculate the radius using the distance formula: [ \text{radius} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ] Given: [ (x_1, y_1) = (7, 9) ] [ (x_2, y_2) = (4, 3) ] [ \text{radius} = \sqrt{(4 - 7)^2 + (3 - 9)^2} ] [ \text{radius} = \sqrt{(-3)^2 + (-6)^2} ] [ \text{radius} = \sqrt{9 + 36} ] [ \text{radius} = \sqrt{45} ] [ \text{radius} = 3\sqrt{5} ]

  2. Calculate the total circumference: [ \text{circumference} = 2\pi \times \text{radius} ] [ \text{circumference} = 2\pi \times 3\sqrt{5} ] [ \text{circumference} = 6\pi\sqrt{5} ]

  3. Calculate the length of the arc for ( \frac{\pi}{4} ) radians: [ \text{arc length} = \text{radius} \times \text{angle in radians} ] [ \text{arc length} = 3\sqrt{5} \times \frac{\pi}{4} ] [ \text{arc length} = \frac{3\pi\sqrt{5}}{4} ]

Therefore, the length of the arc covering ( \frac{\pi}{4} ) radians on the circle is ( \frac{3\pi\sqrt{5}}{4} ).

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Answer from HIX Tutor

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.

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.

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