If a projectile is fired straight upward from the ground with an initial speed of 128 feet per second, then its height h in feet after t seconds is given by the function h(t)equalsnegative 16 t squared plus 128 t. find the maximum height of the projectile.

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4

x = 3 ± √(5+y)