Math in the News: Issue 91

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In this issue of Math in the News we look at applications of math from the Sochi Olympics. Specficially we look at ski jumping and develop a quadratic model based on given data.

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Math in the News: Issue 91

  1. 1. Sochi Math Math in the News Issue 91
  2. 2. #Sochi2014 Math abounds in the Olympics, whether it’s data around Olympic records, medals won, or other statistics. In this issue we will look at the dramatic sport of ski jumping, and its application to quadratics and other non-linear models. Take a look at this interactive to learn more about ski jumping (Source: NY Times). http://www.nytimes.com/newsgraphics/2014/so chi-olympics/ski-jumping.html
  3. 3. #Sochi2014 The path of the ski jumper can be approximated with a parabola. From the NY Times interactive we learn that the skier starts her jump 15 ft off the ground and can cover the length of a football field. We can sketch out a model for this.
  4. 4. #Sochi2014 We get three points: (0, 15), the starting point, (h, k), the vertex of the parabola, and (x, y) where the skier lands. The slant distance covered is 300 ft. and is the hypotenuse of a right triangle.
  5. 5. #Sochi2014 Use the Pythagorean Theorem to find the horizontal distance.
  6. 6. #Sochi2014 You can now find the coordinates of the point where the skier lands.
  7. 7. #Sochi2014 We now have two coordinates and can conceive of the vertex of the parabola, (h, k). From the video we see that the skier elevates by about 3 ft. We can then assign a value to k.
  8. 8. #Sochi2014 The equation for a parabola in vertex form has two variables, a and k, that are unknown. But we do have two coordinates and can solve for these two values.
  9. 9. #Sochi2014 Here is the solution using the first set of coordinates. We end up with an equation that shows a as a function of k.
  10. 10. #Sochi2014 Here is the solution using the second set of coordinates. We end up with a second equation that shows a as a function of k.
  11. 11. #Sochi2014 Substitute one equation into the other to solve for h.
  12. 12. #Sochi2014 Substitute the value of h into one of the equations to find a. We now have all the parameters for the parabola in vertex form.
  13. 13. #Sochi2014 Here is the graph of the parabola, with two key coordinates shown.
  14. 14. #Sochi2014 Now that you’ve seen how to construct a quadratic model using curve-fitting techniques, expand on your work: • Try different values for k to see what impact it has on how far the ski jumper jumps. • Try a longer slant height to see what the equation is for different values of k.

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