When a baseball is hit, its height h (in meters) above the ground t seconds after being hit is given byht=−5t2+3​0t+1

Break Down the Problem

1. Identify the function describing the height of the baseball.
2. Analyze the given quadratic equation: $h(t) = -5t^2 + 30t + 1$.
3. Determine properties such as maximum height and the time at which it occurs.

Relevant Concepts

1. The height function $h(t)$ is a quadratic function of the form $ax^2 + bx + c$.
2. The maximum value of a quadratic function occurs at $t = -\frac{b}{2a}$.

Analysis and Detail

1. In the given function, $a = -5$ and $b = 30$.

2. Calculate the time $t$ at which the maximum height occurs:

   $t = -\frac{b}{2a} = -\frac{30}{2 \cdot -5} = \frac{30}{10} = 3 \text{ seconds}$

3. Substitute $t = 3$ back into the height function to find the maximum height:

   $h(3) = -5(3^2) + 30(3) + 1 = -5(9) + 90 + 1 = -45 + 90 + 1 = 46 \text{ meters}$

Verify and Summarize

• The calculations show that the maximum height of the baseball occurs at $t = 3$ seconds, and the height at this time is $46$ meters.

Final Answer

The maximum height of the baseball is $46$ meters, occurring $3$ seconds after being hit.