A person 6' tall wants to buy a jump rope. If , when the rope is at its lowest point, his hands will be 2' apart and 3' above the ground, and if the rope will take on the shape of a parabola just barely hitting the ground, how long must the rope be? [Consider the form of the rope to be described by the equation , with the origin being a spot on the ground underneath the jumper.]
A) 6.75 feet
B) 6.50 feet
C) 6.25 feet
D) 6.00 feet

Question

A person 6' tall wants to buy a jump rope. If , when the rope is at its lowest point, his hands will be 2' apart and 3' above the ground, and if the rope will take on the shape of a parabola just barely hitting the ground, how long must the rope be? [Consider the form of the rope to be described by the equation  , with the origin being a spot on the ground underneath the jumper.]
A) 6.75 feet 
B) 6.50 feet 
C) 6.25 feet 
D) 6.00 feet

Quizplus · Accepted Answer

Since the hands will be 2 feet apart and 3 feet above the ground, the vertex of the parabolic shape of the rope will be at (1,3). Also, since the rope just barely hits the ground, the y-intercept of the parabola will be 0.

Therefore, the equation of the parabola can be written as:

y = a(x-1)^2 + 3

To find the value of a, we need to use the fact that the rope just barely hits the ground. Since the origin is (0,0), we can substitute these values in the equation and solve for a as follows:

0 = a(0-1)^2 + 3
a = -3

Substituting this value of a into the equation of the parabola, we get:

y = -3(x-1)^2 + 3

Now, we need to find the length of the rope, which is the distance between the two points where the parabola intersects the x-axis. These points can be found by setting y = 0 in the equation of the parabola:

0 = -3(x-1)^2 + 3
(x-1)^2 = 1
x = 0 or x = 2

Since the hands are 2 feet apart, we only need to consider the positive value of x, which is x = 2. Therefore, the length of the rope is:

Length = 2 + 1 = 3 feet

However, the question asks for the length in feet and inches, so we need to convert 3 feet to inches and subtract the extra 6 inches (one for each hand):

Length = 3*12 - 6 = 30 - 6 = 24 inches = 2 feet

Therefore, the length of the rope is 2 + 0.5 = 2.5 feet longer than the height of the person, which is 6 feet.

Total length of the rope = 6 + 2.5 = 8.5 feet = 6.5 feet (to the nearest tenth)

Hence, the answer is B) 6.50 feet.

A Person 6' Tall Wants to Buy a Jump Rope