Question

All ball is thrown into the air with an upward velocity of 24 ft/s. Its height h in feet after t seconds is given by the function h=-16t^2+24t+7. a. In how many seconds does the ball reach its maximum height? Round to the nearest hundredth if necessary. b. What is the ball's maximum height?

Answer 1

The ball reaches its maximum height as your function h(t) is maximized.
h(t) = -16t^2 + 24t + 7
The function opens downward (negative leading coefficient), so it has a high-point / maximum value.

There are \(many\) ways to determine the maximum of a quadratic function. We could... complete the square, use 't=-b/2a' of 'at^2 + bt + c = 0,' or use derivatives & horizontal tangents if you know them.

I think that using \(\displaystyle t = \frac{\neg b}{2a}\) is easiest; \(a = \neg 16\) and \(b = 24\). We simply plug these values in, simplify for the \(t\) value for the time, and then insert this value into \(h(t)\) for the height at t.