If a projectile is fired straight upward from the ground with an initial speed of 64 feet per second, then its height (h) in feet after (t) seconds is given by the function h(t)=-16^2+64t. Find the maximum height of the projectile.

To find the maximum height of the projectile, we need to determine the vertex of the quadratic function \( h(t) = -16t^2 + 64t \). 

### Step 1: Identify the coefficients
The quadratic function is in the form \( h(t) = at^2 + bt + c \), where:
- \( a = -16 \)
- \( b = 64 \)
- \( c = 0 \) (since there is no constant term).

### Step 2: Find the time at which the maximum height occurs
The time at which the maximum height occurs can be found using the formula for the vertex of a parabola:
\[
t = -\frac{b}{2a}
\]
Substituting the values of \( a \) and \( b \):
\[
t = -\frac{64}{2(-16)} = \frac{64}{32} = 2 \text{ seconds}
\]

### Step 3: Calculate the maximum height
Now, substitute \( t = 2 \) seconds back into the height function \( h(t) \) to find the maximum height:
\[
h(2) = -16(2)^2 + 64(2)
\]
\[
h(2) = -16(4) + 128 = -64 + 128 = 64 \text{ feet}
\]

### Conclusion
The maximum height of the projectile is **64 feet**.

Would you like further details or have any questions?

Here are 5 related questions you might be interested in:
1. How do we derive the formula for the vertex of a parabola?
2. What would the height be at \( t = 1 \) second?
3. How long will it take for the projectile to hit the ground?
4. What is the significance of the coefficient \( a \) in the quadratic function?
5. How would the problem change if the initial velocity were different?

**Tip:** The vertex form of a quadratic function provides an easy way to identify the maximum or minimum value of the function.

If a projectile is fired straight upward from the ground with an initial speed of 64 feet per second, then its height (h) in feet after (t) seconds is given by the function h(t)=-16t^2+64t. Find the maximum height of the projectile.

Discover how to find the maximum height of a projectile launched upward with an initial speed of 64 feet per second using the quadratic function h(t) = -16t^2 + 64t. This detailed guide walks you through the vertex formula to calculate the maximum height, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation