Expert: Josh - 6/4/2006

Question
Hi, I'm really stuck on this question-please help!

The path traveled by a golf ball hit with a 9-iron can be modeled with the quadratic function, y= -0.042x^2+5x, where x is the distance in yards from the point it was hit and y is the height of the golf ball in feet. Assume that the ground is level.
A. find the maximum height reached by the ball
B. how far from where it was hit does the ball hit the ground?

thank u sooooooooo much for your help!(p.s. please show me how you got the answer) :)

Answer
Hi Hannah,

Here is what you need to know.

When given a function y=f(x) (like the quadratic in this question), to find the maximum value attained by f(x), you
(i) differentiate y=f(x) with respect to x,
(ii) set this to zero, and
(iii) solve for x.

A. Given y=-0.042x^2+5x,
(i) The first derivative is dy/dx=2*(-0.042)x+5.
(ii) To find the stationary point (i.e., the x-value where the parabola has maximum height), we set dy/dx=0. i.e., 2*(-0.042)x+5=0 ......[#1]
(iii) Now, solving this for x, from [#1], we get
-0.084x+5=0   =>  x=5/0.084.
This tells us that the maximum height of the golf ball's trajectory occurs when x is approximately  xo=59.523 yards.

Note carefully that this x-value is the horizontal distance from the point where the ball was hit. The actual height is given by substituting xo=59.523 into the function y=f(x).

Max. height = f(xo)
= -0.042*(59.523)^2+5*(59.523)
= 148.809 yards.

B. Draw a diagram of the parabola. The trajectory is clearly symmetrical about x=xo. So, the ball must hit the ground at x = 2*xo = 119.047 yards.

You can solve this algebraically, considering the x-intercepts of y=-0.042x^2+5x.
We can factorize this as y=x(-0.042x+5).
Setting the height y=0 (when the ball hits the ground, it must have zero elevation), we get x(-0.042x+5)=0.

One trivial solution is x=0, which tells us nothing. Since we already know that the ball was on the ground when it was hit initially. The useful piece of information is given by setting (0.042x+5)=0. This gives x=5/0.042=119.047. Same answer as before.

Learn the principles well :)