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The problem asks to find the maximum height of a potato launched by a potato launcher using the equation:

\[
h(t) = -16t^2 + 48t + 8
\]

Here, \( h(t) \) is the height in feet, and \( t \) is the time in seconds. The quadratic equation represents a parabolic motion, and since the coefficient of \( t^2 \) is negative, the parabola opens downward, meaning the vertex represents the maximum height.

### To find the maximum height:
1. **Find the time at which the maximum height occurs**:
   The time \( t \) at the maximum height can be found using the formula for the vertex of a parabola:

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

   Here, \( a = -16 \) and \( b = 48 \), so:

   \[
   t = \frac{-48}{2(-16)} = \frac{-48}{-32} = 1.5 \text{ seconds}
   \]

2. **Find the maximum height**:
   Substitute \( t = 1.5 \) back into the original equation to get the maximum height:

   \[
   h(1.5) = -16(1.5)^2 + 48(1.5) + 8
   \]

   Let's compute this.

\[
h(1.5) = -16(2.25) + 72 + 8 = -36 + 72 + 8 = 44 \text{ feet}
\]

Thus, the maximum height of the potato is **44 feet**, and it is reached after **1.5 seconds**.

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Would you like further details or have any questions?

Here are 5 related questions to expand the topic:
1. How would the maximum height change if the initial velocity was different?
2. What would happen if the equation had a different initial height (different constant term)?
3. How can you determine when the potato hits the ground using this equation?
4. How does the acceleration due to gravity appear in this equation?
5. How can we derive the time at which the potato reaches half its maximum height?

**Tip:** For any quadratic function, the vertex represents either the maximum or minimum value, depending on the sign of the quadratic term.

The height h, in feet, of a potato launched from a potato launcher straight up into the air is modeled by the equation h = -16t^2 + 48t + 8, where t is the time in seconds after the potato was launched. What is the maximum height of the potato and when did it reach that height?

Learn how to find the maximum height of a potato launched by a potato launcher using the quadratic equation h = -16t^2 + 48t + 8. This problem covers key algebraic concepts such as quadratic functions, parabolas, and the vertex formula. Suitable for students in Grades 9-12.

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