Intersect of Line and Circle

[Chaim]

Samatha is running near the Circular Park, the shape of a perfect circle. It has a radius of 8 cm. She begins from a point 10 cm west and 3 cm south of the center of the park. She heads toward a point 20 cm east and 4 cm north of the center of the park. Though, when she reaches a point due east of the center of the forest, she turns and runs due south until she exits the park.

Samatha runs at a constant 5 cm per hour. How much time did she spend in the forest?

So the things I bolded were the main given facts and numbers
So I tried drawing a coordinate plane.
1) I made the center of the forest (0,0)
2) She begins from point 10 cm west and 3 cm south, which means she began at (-10, -3)
3) She heads toward a point 20 cm east and 4 cm north of the center of the forest, so it's at (20,4), but she's heading there
4) I don't know if I'm correct here, she moves east now to the center of the forest then runs south?
Then (5) would be another step,
Would I use the standard equation of the circle? x²+y²=r², r being the radius

If someone could help me by providing the steps or explaination, that would be nice
Thanks!

 

[soroban]

Hello, Chaim!

Are you sure about those units?

The problem mixes kilometers and centimeters,
. . which makes the solution very ugly.

Is that intentional?

 

[Chaim]

Hello, Chaim!

Are you sure about those units?

The problem mixes kilometers and centimeters,
. . which makes the solution very ugly.

Is that intentional?

OH OOPS!
I accidently was typing it and accidently did some typos!
Yeah sorry, it's all cm

 

[soroban]

Hello, Chaim!

I believe the units are kilometers.
. . (A circular park with a radius of 8 centimeters is less than 7 inches across.)
Also, you refer to a park . . . and then a forest.

I'll outline a game plan for this problem.

Samatha is running near Circular Park, the shape of a perfect circle with a radius 8 km.
She begins from a point 10 km west and 3 km south of the center of the park.
She heads directly toward a point 20 km east and 4 km north of the center.
When she reaches a point due east of the center, she turns and runs due south and exits the park.
Samatha runs at a constant 5 km per hour. .How much time did she spend in the park?

                        |
                      * * *                         (20,4)
                  *     |     *                        o B
                *       |       *                 o    :
               *        |        *           o         :
                        |               o              :
              *         |     Q   *o                   :
      - - + - * - - - - * - - o - * - - - - - - - - - -+- -
          :   *         |o    :   *
          :    P    o   |     :
          :    o        |     :  *
        A o     *       |     : *
      (-10,-3)    *     |     o
                      * * *   R
                        |

The equation of the circular park is: .\(\displaystyle x^2 + y^2 \:=\:64\)

Samantha starts at \(\displaystyle A(\text{-}10,\text{-}3)\) and runs to \(\displaystyle B(20,4).\)
The equation of her line is: .\(\displaystyle y \:=\:\frac{7}{30}x - \frac{2}{3}\)

She first enters the park at \(\displaystyle P\).
We must find the first intersection of the line and the circle.

She cross the x-axis at \(\displaystyle Q.\)
We must find the x-intercept, \(\displaystyle x_o\), of the line.

She runs directly south and exits at \(\displaystyle R.\)
We must find the coordinates of \(\displaystyle R.\)
We know the x-coordinate is \(\displaystyle x_o.\)
We can find the y-coordinate on the circle.

Her total distance in the park is: .\(\displaystyle PQ + QR.\)
. . (Use the Distance Formula.)

Then divide by her speed, 5 km/hr.

This give us her total time spent in the park.


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