Question

A jump rope is being spun by Kimmy and Jimmy holding it at the same height. At the top of each spin, the rope is in a parabolic form modeled by the equation

h(x)=-(0.1598x)(x-11) where h is the height of the rope off the ground and x is the horizontal distance from a tree behind Jimmy.
By how much would the rope clear the top of the head of a 5-foot-tall child, or how much would they need to duck to dodge the rope?

Answer 1

To find out by how much a 5-foot-tall child would need to duck, we calculate the maximum height of the rope at the vertex of the parabola. The rope reaches a maximum height of about 4.756 feet, therefore the child would need to duck by approximately 2.93 inches to dodge the rope.

Step-by-step explanation:

The jump rope's height above the ground at any point x is given by the parabolic equation h(x) = -(0.1598x)(x-11). To determine how much a 5-foot-tall child would need to duck or by how much the rope would clear the top of the child's head, we need to find the maximum height of the rope when it is spun. The vertex of a parabola given by the equation y = ax^2 + bx + c is located at x = -b/(2a), which represents the horizontal distance from a reference point at which the maximum height is attained.

In this case, a = -0.1598 and b = 0.1598*11. Plugging these into the vertex formula gives us x = -(-0.1598*11)/(2*(-0.1598)) = 5.5. Substituting x = 5.5 back into the original equation, we find the maximum height of the rope: h(5.5) = -(0.1598*5.5)(5.5-11). After calculating, the result is h(5.5) = 4.756 feet.

Since the child is 5 feet tall, the rope would need to clear 5 feet to avoid hitting the child. Therefore, the child would need to duck 5 - 4.756 = 0.244 feet, or approximately 2.93 inches, to dodge the rope.