h(x)=-0.005x^2+4.2x+900 Word Problem

Question

A motorcycle stunt rider jumped over the canyon. The path of her motorcycle was given approximately by the function H(x)=−0.005x^2+4.2x+900, where H is measured in feet above the ground and x is the horizontal distance from the launch ramp. 

A) How high above the ground was the launch ramp? [Hint: what is the value of x if she hasn't jumped yet?] 

B) What was the rider’s maximum height above the ground, and how far horizontally was the rider from the ramp when she reached the maximum height? Show your answer using two different techniques – one graphically and one algebraically. 

C) On the other side of the river, after reaching maximum height, is a landing platform that is H=800 above the ground on another cliff. How far did the rider travel across? Show your answer both graphically and algebraically. Round your answer to the nearest foot. 

(No calculus)

Answer 1

Hello, Ximena,

A) The height of the launch ramp can be found by using the equation (H(x)=−0.005x^2+4.2x+900) and setting x, the time, to 0, the start of her jump.

H(x)=−0.005(0)^2+4.2(0)+900

H(x) = 900 feet (height of the launch ramp)

B) What was the rider’s maximum height above the ground, and how far horizontally was the rider from the ramp when she reached the maximum height?

1. Graphical: Plot the equation H(x)=−0.005x^2+4.2x+900 You'll find it is a parabola with a vertex of (420,1782). At 420 feet from the launch ramp, she is 1782 feet high.
2. Algebraic: The question states "no calculus." This would certainly offend Issac Newton, and I intend to let him know. The actual problem is that I don't know how to do this algebraically, other than to make a table of distances from the launch and calculate the height from the equation until the height begins to sdecrease. An Excel spreadsheet simplifies this, but it is still ugly.

[Spoiler] The first derivative would have given us the slope of the line as a function of distance. The point at which the jumper is a maximum height will have a slope of zero, the point at which it changes direction. If we set the first derivative to zero and solve for distance, we will find the distance to maximum height. The first derivative is: H'(x)=−0.010x+4.2. Set it equal to zero and solve for x. I find x = 420. The height is maximum at 420 feet from the launch.

C) How far did the rider travel across? Take the height equation and solve for x when H(x) is 800: H(x)=−0.005x^2+4.2x+900

800=−0.005x^2+4.2x+900

0 = −0.005x^2+4.2x+800

Use the phthagorean theorem to solve. There are two roots, -23.17 and 863.2. I pick 863, since the -23 would mean she fell off the lauch ramp backwards. Not good for the reputation. She travelled 863 feet to reach to other side at 800 feet elevation.

Bob