Results 1 to 9 of 9

Math Help - Intersect of Line and Circle

  1. #1
    Member
    Joined
    Oct 2011
    Posts
    153

    Intersect of Line and Circle

    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!
    Last edited by Chaim; April 9th 2012 at 04:54 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,674
    Thanks
    445

    Re: Intersect of Line and Circle

    Quote Originally Posted by Chaim View Post
    Samatha is running near the Circular Park, the shape of a perfect circle. It has a radius of 8 km. 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 km 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 km south, which means she began at (10, -3)
    3) She heads toward a point 20 cm east and 4 km 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!
    you do realize that 5 cm/hr is centimeters per hour ? a snail moves faster.

    sure you don't mean km (kilometers) instead of cm (centimeters) everywhere in this problem?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Nov 2007
    From
    Trumbull Ct
    Posts
    910
    Thanks
    27

    Re: Intersect of Line and Circle

    check distances and rate of running. cm per hour ????
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Oct 2011
    Posts
    153

    Re: Intersect of Line and Circle

    Ah, it's just the units
    But yeah, I accidently messed up, it's all suppose to be the same units, so use centimeters xD
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,674
    Thanks
    445

    Re: Intersect of Line and Circle

    she starts at (-10,-3) , not (10,3)

    she heads toward the point (20,4)

    start by finding the equation of the line between (-10,-3) and (20,4)

    once you find that linear equation, solve for either x or y (whichever is easier) and substitute the result into the circle equation, then solve for the single variable ... understand that the line first intersects the circle in quadrant III (x and y both negative).

    you also need to determine where the line crosses the x-axis (x-intercept), because that is where she turns to head due south.

    attached is a sketch of the problem ... the red line segments mark the path she travels.

    ... one final tip, use kilometers for distance.
    Attached Thumbnails Attached Thumbnails Intersect of Line and Circle-circleprob.jpg  
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Oct 2011
    Posts
    153

    Re: Intersect of Line and Circle

    Quote Originally Posted by skeeter View Post
    she starts at (-10,-3) , not (10,3)

    she heads toward the point (20,4)

    start by finding the equation of the line between (-10,-3) and (20,4)

    once you find that linear equation, solve for either x or y (whichever is easier) and substitute the result into the circle equation, then solve for the single variable ... understand that the line first intersects the circle in quadrant III (x and y both negative).

    you also need to determine where the line crosses the x-axis (x-intercept), because that is where she turns to head due south.

    attached is a sketch of the problem ... the red line segments mark the path she travels.

    ... one final tip, use kilometers for distance.
    Lol okay I'll use kilometers
    And oopsies, yeah, I keep making typos xD, so yeah it's (-10,-3)
    Ooo I see! So basically I use y=mx+b right?
    Using (-10, -3) and (20,4) since she's making a straight line there?
    So (4-(-3))/(20-(-10))=(7/30)
    Then y=(7/30)x+b
    So using (-10, -3) to find b
    -3=(7/30)(10)+b
    -3=7/3+b
    -9/3=7/3+b
    -16/3=b
    So y=(7/30)x-(16/3)
    Then to find y intercept -> y=(7/30)(0)-(16/3) -> y=-16/3
    Then to find x-intercept -> 0=(7/30)x-(16/3)-> 16/3=(7/30)x -> x = 112/90
    Just making sure, am I doing this right so far?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,735
    Thanks
    642

    Re: Intersect of Line and Circle

    Hello, Chaim!

    I will assume that all the measurements are in kilometers.
    I'll outline the Game Plan for this problem.


    Samatha is running through Circular Park, in the shape of a perfect circle with a radius of 8 km.
    She begins from a point 10 km west and 3 km south of the center of the park.
    She heads toward a point 20 km east and 4 km north of the center of the park.
    When she reaches a point due east of the center of the park,
    . . she turns and runs due south until she exits the park.

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

    Code:
                            |
                          * * *                          (20.4)
                      *     |     *                         o B
                    *       |       *                 *     : 
                   *        |        *          *           :
                            |               *               :
                  *         |   Q     *                     :
    - - * - - - - * - - - - + - o - - * - - - - - - - - - - * - - 
        :         *       * |   |     *
        :       P o *       |   |
        :     *    *        |   |    *
      A o           *       |   |   *
    (-10,-3)          *     |   o *
                          * * *
                            |   R
    The equation of the park is: . x^2 + y^2 \,=\,64

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

    She enters the park at point P.
    We must find the first intersection of line AB and the circle.

    She cross the x-axis at point Q.
    We must find the x-intercept, x_o, of line AB.

    Then she runs directly south and exits the park at point R.
    We must determine y (in the circle) when x = x_o


    Then we find her total distance in the park, PQ + QR km,
    . . and divide by her speed, 5 km/hr.

    This gives us her time spent in the park.



    Ah! . . . Skeeter already explained all this . . . *sigh*
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Super Member
    Joined
    Nov 2007
    From
    Trumbull Ct
    Posts
    910
    Thanks
    27

    Re: Intersect of Line and Circle

    Once you get the equation as shown by Soroban the two running distances are easily solved by PYt theorem .The turning point where y=0 is 60/21
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member
    Joined
    Oct 2011
    Posts
    153

    Re: Intersect of Line and Circle

    Quote Originally Posted by Soroban View Post
    Hello, Chaim!

    I will assume that all the measurements are in kilometers.
    I'll outline the Game Plan for this problem.



    Code:
                            |
                          * * *                          (20.4)
                      *     |     *                         o B
                    *       |       *                 *     : 
                   *        |        *          *           :
                            |               *               :
                  *         |   Q     *                     :
    - - * - - - - * - - - - + - o - - * - - - - - - - - - - * - - 
        :         *       * |   |     *
        :       P o *       |   |
        :     *    *        |   |    *
      A o           *       |   |   *
    (-10,-3)          *     |   o *
                          * * *
                            |   R
    The equation of the park is: . x^2 + y^2 \,=\,64

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

    She enters the park at point P.
    We must find the first intersection of line AB and the circle.

    She cross the x-axis at point Q.
    We must find the x-intercept, x_o, of line AB.

    Then she runs directly south and exits the park at point R.
    We must determine y (in the circle) when x = x_o


    Then we find her total distance in the park, PQ + QR km,
    . . and divide by her speed, 5 km/hr.

    This gives us her time spent in the park.



    Ah! . . . Skeeter already explained all this . . . *sigh*
    Ah.. I see! Ok thanks!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Line plane intersect within boundaries
    Posted in the Geometry Forum
    Replies: 4
    Last Post: November 18th 2010, 02:35 AM
  2. Replies: 2
    Last Post: April 26th 2009, 02:23 AM
  3. Replies: 2
    Last Post: December 9th 2008, 01:41 AM
  4. Finding planes which intersect at a given line
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: June 8th 2008, 10:39 PM
  5. Intersect Circle vs Line vs Speed
    Posted in the Geometry Forum
    Replies: 0
    Last Post: March 29th 2007, 02:58 AM

Search Tags


/mathhelpforum @mathhelpforum