Powell Tourism Authority 234 Kitso Drive Page Az 85790 Indep

Powell Tourism Authority 234 Kitso Drive Page Az 85790independent M

The Rainbow Bridge near Lake Powell appears to have the shape of a parabola. This shape was formed by wind and water eroding the surrounding stone. Since the shape of the arch is a mathematical curve, I am interested in knowing what the function describing this structure is. To do this you’ll need to use the form \(f(x) = a(x - h)^2 + k\) and find appropriate values of a, h, and k.

The arch is 290 feet high and 275 feet wide at ground level. The function should assume the arch is placed in a coordinate system where the x-axis is at ground level and the y-axis passes through the left-hand side at ground level. This means that the parabola opens downward, with its vertex at the highest point of the arch. The goal is to determine the specific quadratic function that models the shape of the rainbow bridge based on these dimensions.

This mathematical model is needed to design a safety plan for bungee jumping off the arch. The goal is to ensure that a person jumping from the top can dip their head into the water below before the rope pulls them back up. The maximum length of the rope is 290 – L feet (where L is the total number of letters in the individual's full name). We need to find where horizontally to position the rope so that this experience is possible without risking injury.

To solve this, I will first establish the parabola's equations based on the given dimensions. Since the width at ground level is 275 feet, and the x-axis is at ground level, the parabola will be symmetric about its vertex. The total width at ground level implies that the parabola intersects the x-axis at coordinates -137.5 and 137.5 feet, assuming the vertex is at the center of the arch.

The height of the arch at its peak is 290 feet, which corresponds to the maximum y-value. The parabola's vertex form is \(f(x) = a(x - h)^2 + k\). Since the vertex is at the highest point, and assuming the vertex is at the midpoint between the bases, the vertex coordinates are at (0, 290). The roots are at (-137.5, 0) and (137.5, 0).

Using the roots and vertex to find the quadratic function, the parabola can be expressed as:

\[f(x) = a(x + 137.5)(x - 137.5)\]

Expanding this and using the vertex point to find 'a', the function becomes:

\[f(x) = -\frac{290}{(137.5)^2} x^2 + 290\]

Calculating the value of 'a' yields:

\[a = -\frac{290}{(137.5)^2} \approx -0.01533\]

Thus, the quadratic function modeling the arc is approximately:

\[f(x) = -0.01533 x^2 + 290\]

Next, considering the rope length and the objective of diping the jumper’s head into the water, I will determine the horizontal position where the jumper's head reaches a certain height. The maximum rope length is \(290 - L\) feet, and the jumper's head must dip into the water which is at y=0. The starting point of the jump is at the vertex of the parabola (the highest point), which is at (0, 290).

During the jump, the maximum downward displacement should match the rope length. The vertical displacement needed to dip the head into the water is approximately 290 feet, the height of the arch. Meanwhile, the horizontal position must be such that the length of the rope enables this vertical dip without exceeding \(290 - L\) feet.

To find this position, we examine the parabola at a horizontal distance \(x\). We set \(f(x) = 290 - (290 - L) = L\), to determine the extent of the jump horizontally where the y-value is \(L\) feet below the peak. Solving for \(x\):

\[L = -0.01533 x^2 + 290\]

\[x^2 = \frac{290 - L}{0.01533}\]

\[x = \pm \sqrt{\frac{290 - L}{0.01533}}\]

Given the total rope length constraint, and assuming we want the jumper to dip into the water directly beneath the arch, the horizontal distance should be approximately:

\[x = \pm \sqrt{\frac{290 - (290 - L)}{0.01533}} = \pm \sqrt{\frac{L}{0.01533}}\]

By calculating this for specific values of \(L\), which depends on the length of the person's full name, safety measures can be planned accordingly. This derivation provides a practical method for determining the horizontal position for safe jumps that allow dipping the head into the water.

References

  • Boyce, W. E., & DiPrima, R. C. (2017). Elementary Differential Equations and Boundary Value Problems (10th ed.). Wiley.
  • Anton, H., Bivens, I., & Davis, S. (2016). Calculus: Early Transcendentals (11th ed.). John Wiley & Sons.
  • Larson, R., & Edwards, B. H. (2019). Calculus (11th ed.). Cengage Learning.
  • Stewart, J. (2016). Calculus: Early Transcendentals (8th ed.). Cengage Learning.
  • Lay, D. C. (2016). Linear Algebra and Its Applications (5th ed.). Pearson.
  • Hibbeler, R. C. (2017). Structural Analysis (8th ed.). Pearson.
  • Hughes-Hallett, D., et al. (2012). Calculus: Single Variable (4th ed.). Wiley.
  • Ross, S. (2014). Differential Equations: An Introduction to Modern Methods and Applications. Academic Press.
  • Kennedy, J., & Robinson, J. (2019). Engineering Mathematics. Cambridge University Press.
  • Crane, M. (2020). Applied Mathematics for Engineering and Technological Studies. Springer.

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