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The problem gives us the height \( h(t) \) of a baseball, where \( t \) is the time in seconds after it is hit. The height function is:

\[
h(t) = -4.9t^2 + 34.3t + 1
\]

### 1. Time to Reach Maximum Height:
To find the time it takes to reach the maximum height, we can use the fact that the maximum of a quadratic function \( h(t) = at^2 + bt + c \) occurs at:

\[
t = \frac{-b}{2a}
\]

Here, \( a = -4.9 \) and \( b = 34.3 \). Plugging in these values:

\[
t = \frac{-34.3}{2 \times -4.9} = \frac{-34.3}{-9.8} = 3.5 \, \text{seconds}
\]

So, it takes **3.5 seconds** for the baseball to reach its maximum height.

### 2. Maximum Height:
Now, substitute \( t = 3.5 \) into the height function \( h(t) \):

\[
h(3.5) = -4.9(3.5)^2 + 34.3(3.5) + 1
\]
\[
h(3.5) = -4.9(12.25) + 34.3(3.5) + 1
\]
\[
h(3.5) = -60.025 + 120.05 + 1
\]
\[
h(3.5) = 61.025 \, \text{meters}
\]

So, the maximum height reached by the baseball is **61.0 meters** (rounded to one decimal place).

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Would you like further details or have any other questions about the steps?

Here are 5 related questions to consider:
1. How would the problem change if the initial velocity was different?
2. What happens to the maximum height if we change the gravitational constant?
3. Can we solve for the time when the baseball returns to the ground?
4. How can we model the trajectory of the baseball with air resistance included?
5. How would you determine the velocity of the baseball at its maximum height?

**Tip**: The vertex form of a quadratic equation is helpful for quickly determining maxima or minima in projectile motion problems!

A baseball player swings and hits a pop fly straight up in the air to the catcher. The height of the baseball in meters t seconds after it is hit is given by the quadratic function h(t) = -4.9t^2 + 34.3t + 1. How long does it take for the baseball to reach its maximum height? What is the maximum height obtained by the baseball?

Solve a quadratic function problem to find the time and maximum height of a baseball hit straight up. Learn how to apply the formula for the vertex of a parabola to projectile motion.

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