Question: A ball is thrown up on the surface of a moon. Its height above the lunar surface (in feet) after $t$ seconds is given by the formula
$$h = 324t - \frac{12}{3} t^2.$$
Find the time that the ball reaches its maximum height.
Answer = (Round to two decimal places as needed.)
Find the maximal height attained by the ball
Answer = (Round to two decimal places as needed.)
1. **State the problem:** We are given the height function of a ball thrown on the moon as $$h(t) = 324t - \frac{12}{3} t^2.$$ We need to find the time $t$ when the ball reaches its maximum height and then find that maximum height.
2. **Simplify the formula:** Simplify the coefficient of $t^2$:
$$h(t) = 324t - 4t^2$$
3. **Identify the type of function:** This is a quadratic function of the form $$h(t) = at^2 + bt + c$$ where $$a = -4,$$ $$b = 324,$$ and $$c = 0.$$ Since $$a < 0,$$ the parabola opens downward, so the vertex represents the maximum point.
4. **Find the time at maximum height:** The time at which the maximum height occurs is given by the vertex formula:
$$t = -\frac{b}{2a} = -\frac{324}{2 \times (-4)} = -\frac{324}{-8} = 40.5$$ seconds.
5. **Calculate the maximum height:** Substitute $$t = 40.5$$ back into the height function:
$$h(40.5) = 324(40.5) - 4(40.5)^2$$
Calculate each term:
$$324 \times 40.5 = 13122$$
$$4 \times (40.5)^2 = 4 \times 1640.25 = 6561$$
So,
$$h(40.5) = 13122 - 6561 = 6561$$ feet.
6. **Final answers:**
- Time to reach maximum height: $$40.50$$ seconds (rounded to two decimal places).
- Maximum height attained: $$6561.00$$ feet (rounded to two decimal places).