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The path of a skateboarder can be modeled by the equation $(x-3)^2=-8(y-8)$, where distance is in feet and air resistance is insignificant. If the skateboarder makes a jump from a ramp at $x=0$, what is her maximum height and how far is she horizontally from the end of the ramp when she lands?


AMaximum height 3 feet; horizontal distance 16 feet
BMaximum height 7 feet; horizontal distance 11 feet
CMaximum height 8 feet; horizontal distance 11 feet
DMaximum height 8 feet; horizontal distance 16 feet

Verified Answer :

  • Identify the vertex \((3,8)\) of the parabola \((x-3)^2 = -8(y-8)\) to find the maximum height of 8 feet.
  • Determine the horizontal distance by solving for \(x\) at ground level (\(y=0\)): \((x-3)^2 = 64\), leading to \(x=11\).
  • Calculate the horizontal distance from the jump start at \(x=0\) to the landing point \(x=11\), which is 11 feet.
  • The maximum height is 8 feet, and the horizontal distance is 11 feet. \(\boxed{8 \text{ feet; } 11 \text{ feet}}\)

Explanation

  1. Determining the Maximum Height

The motion of the skateboarder is modeled by the parabolic equation \((x-3)^2 = -8(y-8)\). This equation is in the form \((x-h)^2 = 4p(y-k)\), where \((h, k)\) is the vertex of the parabola. Identifying the vertex from our equation gives \((h, k) = (3, 8)\). Since the parabola opens downwards (\(4p = -8 < 0\)), the maximum height attained by the skateboarder is the y-coordinate of the vertex, which is \(8\) feet.

  1. Calculating the Landing Distance

To find the landing point, we assume the skateboarder lands at ground level, which corresponds to \(y = 0\). Substituting \(y = 0\) into the equation \((x-3)^2 = -8(y-8)\), we have:
\[(x-3)^2 = -8(0-8) = -8(-8) = 64.\]
Taking the square root of both sides gives \(x - 3 = \pm 8\). This results in two values for \(x\):
\[x = 3 + 8 = 11 \quad \text{and} \quad x = 3 - 8 = -5.\]
Since the jump begins at \(x=0\), the landing point occurs at \(x = 11\). The horizontal distance from the start of the jump to the landing point is \(11 - 0 = 11\) feet.

  1. Conclusion

Based on the calculations, the maximum height is \(8\) feet, and the horizontal distance from the start of the ramp at \(x=0\) to the landing point at \(x=11\) is \(11\) feet. This corresponds to the third option. Thus, the skateboarder reaches a maximum height of 8 feet and lands 11 feet away horizontally from the start of her jump. \(\boxed{8 \text{ feet; } 11 \text{ feet}}\)

Examples

This problem is an application of parabolic motion, which is commonly used in physics to analyze the trajectories of projectiles, such as jumping skateboarders, basketball shots, or throwing a ball. By modeling paths with quadratic equations, engineers and athletes can calculate the peak elevation and total range of a jump, ensuring safety and performance in sports facility design or athletic training.

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