1. **State the problem:** We are given the height function of an object in terms of time $t$: $$h = -16t^2 + 60t + 3$$ and we want to understand its behavior. 2. **Identify the type of function:** This is a quadratic function of the form $$h(t) = at^2 + bt + c$$ where $a = -16$, $b = 60$, and $c = 3$. 3. **Important rules:** - The graph of a quadratic function is a parabola. - Since $a = -16 < 0$, the parabola opens downward, meaning it has a maximum point. - The vertex formula for the time $t$ at which the maximum height occurs is $$t = -\frac{b}{2a}$$. 4. **Calculate the time at maximum height:** $$t = -\frac{60}{2 \times (-16)} = -\frac{60}{-32} = \frac{60}{32} = \frac{15}{8} = 1.875$$ seconds. 5. **Calculate the maximum height by substituting $t=1.875$ into $h(t)$:** $$h(1.875) = -16(1.875)^2 + 60(1.875) + 3$$ $$= -16(3.515625) + 112.5 + 3$$ $$= -56.25 + 112.5 + 3$$ $$= 59.25$$ feet. 6. **Interpretation:** The object reaches its maximum height of 59.25 feet at 1.875 seconds. 7. **Additional info:** The initial height at $t=0$ is $h(0) = 3$ feet. **Final answer:** The maximum height is $$\boxed{59.25}$$ feet at $$t = 1.875$$ seconds.