SOLUTION: The path of a diver is given by y= -4/9x^2 + 24/9x +12 where y is the height in feet, and x is the hoizontal distance from the end of the diving board in feet. What is the maximum

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Question 13337: The path of a diver is given by y= -4/9x^2 + 24/9x +12 where y is the height in feet, and x is the hoizontal distance from the end of the diving board in feet. What is the maximum height of the dive?
I tried this on the ti-83 plus and got y= 16 feet, however, looking at the graph, I don't know which x value to use or how to work the problem once I find it. Thank you for your help. I was able to use one of your already worked problems to help me with another one of mine by substituting. This is a very helpful site.
Answer by bam878s(77) (Show Source):
You can put this solution on YOUR website!
Hi.
The diver follows a path that is a upside-down parabola

You were right by saying the maximum point occourse when y=16.
why?
First you must find the x-value that will give a local maxima or minima of the graph.
This is just a Calculus problem
you start by taking the derivative of the function to get
(-8/9)x + (24/9) = y.
to find a local maxima you set y=0 and solve for x
(-8/9)x + (25/9) = 0.
(-8/9)x = (-25/9)
divide both sides by (-8/9 to get
x=3.
You know this will be a local maxima or minimum by this mehtod because you are setting the slope of the derivative (or tangent line to original graph) equal to zero. All tangents after 16 will be negative because the graph is curving downward. Also all tangents before x=3 will be positive because the graph is increasing up until this point.
I hope this helps you and doesn't confuse you.
I don't know about your calculus background.