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Arch Investigation

Arch Investigation. B. Davis MathScience Innovation Center. Do the lights at this school cast parabolas on the wall?. At the top of the stairs indirect lighting casts an interesting pattern on our walls. Is this shape a parabola or most likely another math shape?. Let’s Investigate!.

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Arch Investigation

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  1. Arch Investigation B. Davis MathScience Innovation Center

  2. Do the lights at this school cast parabolas on the wall? At the top of the stairs indirect lighting casts an interesting pattern on our walls. Is this shape a parabola or most likely another math shape? Arch of Light B. Davis MathScience Innovatio Center

  3. Let’s Investigate! • Step one: take some photos • Select a clear photo for the investigation Arch of Light B. Davis MathScience Innovatio Center

  4. Step 2: • We will superimpose our photo on some graph paper • The graph paper will help us to locate points on the curve • We need at least 3 points Arch of Light B. Davis MathScience Innovatio Center

  5. Step 3: • We might adjust the picture to put it in a convenient place. • We must be careful not to distort the picture’s dimensions. • Record the 3 points. Arch of Light B. Davis MathScience Innovatio Center

  6. Step 4: Results • the 3 points: • (1,1) • (5,3) • (13,2) Arch of Light B. Davis MathScience Innovatio Center

  7. Step 4: Write system • Write 3 equations, one for each point:(1,1) (5,3) (13,2) • using quadratic equation: y = ax^2 + bx + c • 1: 1 = a + b + c • 2: 3 = 25a + 5b + c • 3: 2 = 169a + 13 b + c Arch of Light B. Davis MathScience Innovatio Center

  8. Step 5: Change to Matrix Equation 1 = a + b + c 3 = 25a + 5b + c 2 = 169a + 13 b + c a b c 1 1 1 25 5 1 169 13 1 1 3 2 Arch of Light B. Davis MathScience Innovatio Center

  9. 1 1 1 25 5 1 169 13 1 Step 6: Solve Matrix Equation a b c 1 3 2 Inverse matrix is : Solution is: 1/48 - 1/32 1/96 -3/8 7/16 -1/16 65/48 -13/32 5/96 -5/96 13/16 23/96 Arch of Light B. Davis MathScience Innovatio Center

  10. Step 7: Write Math Model a = - 5/96 b = 13/16 c = 23/96 Solution is: Therefore the equation for the light is: y = ax^2 + bx + c y = -5/96x^2 + 13/16x + 23/96 Arch of Light B. Davis MathScience Innovatio Center

  11. Step 8: Regression analysis Enter 3 points into STAT EDIT Use STAT CALC 5 to find line of best fit Therefore the regression equation for the light is: Y= -.05208x^2 + .8125x + .23985 With a correlation coefficient of: 1 Arch of Light B. Davis MathScience Innovatio Center

  12. Final Results The light hitting the wall could very well be a parabolic shape. The 3 points: • (1,1) • (5,3) • (13,2) lie on the line Y= -.05208x^2 + .8125x + .23985 Arch of Light B. Davis MathScience Innovatio Center

  13. Step 9: Comparison and Conclusion When the decimals found using STAT CALC 5 are evaluated using VARS 5 EQ a MATH 1 VARS 5 EQ b MATH 1 VARS 5 EQ c MATH 1, it is found that they exactly match the a,b,c found using the matrix method to solve the system. With r =1, it is concluded that both methods give an accurate method of finding the line of best fit. The light hitting the wall could very well be a parabola. Arch of Light B. Davis MathScience Innovatio Center

  14. Further Note: • Although the conclusion gives strong evidence that this curve is a parabola, further investigation may reveal that it is a hyperbola. The difference is whether or not the wall is exactly parallel to the axis of the cone of light. If it is even slightly off, a hyperbola (rather than a parabola) is the result. Arch of Light B. Davis MathScience Innovatio Center

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