SOLUTION: Hello I have a problem solving this parabola question.The path of a cliff diver as he dives into a lake ,is given by this eqaution,y=-(x-10)(SQAURED)+75,where y metres is the diver

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Question 310688: Hello I have a problem solving this parabola question.The path of a cliff diver as he dives into a lake ,is given by this eqaution,y=-(x-10)(SQAURED)+75,where y metres is the diver's height above the water and,x metres is the horisontal distance travelled by the diver.What is the maximum height the diver is above the water?
Found 2 solutions by Fombitz, solver91311:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
y=-%28x-10%29%5E2%2B75
Since the first term is always negative (since it's squared and multiplied by -1), the max value it reaches is zero (when x=10), which then only leaves the constant term.
ymax=75
.
.
.
+graph%28+300%2C+300%2C+-1%2C+19%2C+-10%2C+90%2C+-%28x-10%29%5E2%2B75%29+

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


Not sure whether you just need an intuitive approach, a formal algebraic approach, or a calculus approach.

Intuitive

The largest y can be is 75 and that is when . That is because is always a positive number, unless it is zero, so is always a negative number unless it is zero. Therefore you are always subtracting something from 75 unless , which happens when

While we are at it -- 75 meters? That is 246 feet. Hitting the water from that height would be like hitting concrete. I rather think your cliff diver is only going to perform this particular stunt once.

Algebraic



Expand the binomial expression:





A parabola with a negative lead coefficient opens downward, hence the vertex is a maximum. The vertex of a parabola expressed in form has a vertex that occurs at an value of , so:



And the value of the function at is:



Calculus

Using the function definition from the Algebraic discussion:



The function will have a maximum where the first derivative is zero and the second derivative is negative.



Set it equal to zero







which is negative for all in the domain of

So there is a maximum at



John