Question 147130
The path of the diver is given by y= 4/9x^2 + 24/9x + 12
where 
y is the height in feet
x is the horizontal distance from the end of the diving board (in feet).
:
The coefficient of x^2 has to be negative to have a maximum, assuming the equation is:
y = {{{-4/9}}}x^2 + {{{24/9}}}x + 12
:
Find the axis of symmetry using the formula: x =-b/(2a); a=-4/9; b=24/9
x = {{{(-24/9)/(2*(-4/9))}}} = {{{(-24/9)/((-8/9))}}}
Invert the dividing fraction and multiply
x = {{{(-24/9)*(-9/8)}}} cancel and you have:
x = +3 ft horizontal distance from end of diving board for max height
:
What is the maximum height of the diver?
:
Substitute 3 for x in the original equation to find max height
y = {{{-4/9}}}3^2 + {{{24/9}}}3 + 12
:
y = {{{-4/9}}}9 + {{{72/9}}} + 12
cancel
y = -4 + 8 + 12
:
y = 16 ft is max height
:
:
A graph of this equation will make it clear:
{{{ graph( 300, 200, -6, 12, -10, 20, (-4/9)x^2+(24/9)x+12) }}}
Note this also shows the diving board to be 12 ft above the water and he will enter the water 9 ft from the end of the diving board.
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Did this make some sense to you?