I need help on this algebra 2 question

Question

A competition is held to see who can throw a baseball the highest from the front row of the upper deck seats at the stadium, 83 ft above field level. The winner throws the ball with an initial vertical velocity of 92 ft/sec and it lands on the infield grass.

 

Using the free-fall motion function, h(t) = -16t2 + v0t + h0, where t is the time in seconds, h(t) is the height above the ground in ft, v0 is the initial vertical velocity in ft/s, and h0 is the initial height in ft, find the highest height the ball reached.

 

Type your answer rounded to the nearest hundredth.

Answer 1

We are given this quadratic function h(t) = -16t² + v₀t + h₀ with v₀ = 92 ft/sec and h₀ = 83 feet.

Hence h(t) = -16t² + 92t + 83

As we want to find the maximum of this function, I would differentiate (with respect to t) to get:

h'(t) = -32t + 92

And -32t + 92 = 0 because the differential is the gradient of the curve, and at a maximum the gradient is zero.

There's only one max/min, so it's the maximum required. Or, if you prefer, differentiate again to get h"(t) = -32. Negative second derivative indicates a maximum.

Therefore t = 92/32 = 2.875 seconds

Substitute this time value in the original equation to get:

h(t) = -16(2.875)² + 92(2.875) + 83 = 215.25 feet

You can double-check with Desmos Graphing or similar.