A projectile is fired straight up from ground level. After t​ seconds, its height above the ground is s​ feet, where s=−16t2+112t.
For what time period is the projectile at least 160 feet above the​ ground?

We are given the height function of the projectile:

\[
s(t) = -16t^2 + 112t
\]

We are asked to find for what time period the projectile is at least 160 feet above the ground, meaning we need to solve for \( t \) where:

\[
s(t) \geq 160
\]

Substitute \( s(t) = 160 \) into the equation:

\[
-16t^2 + 112t = 160
\]

Now let's solve this inequality step by step.

### Step 1: Set the equation to 0
Move everything to one side of the equation:

\[
-16t^2 + 112t - 160 = 0
\]

### Step 2: Divide through by \(-16\) to simplify

\[
t^2 - 7t + 10 = 0
\]

### Step 3: Solve the quadratic equation
We can factor this quadratic:

\[
(t - 5)(t - 2) = 0
\]

The solutions are:

\[
t = 5 \quad \text{and} \quad t = 2
\]

### Step 4: Interpret the result
The projectile reaches exactly 160 feet at \( t = 2 \) seconds when going up and \( t = 5 \) seconds when coming down. Thus, the projectile is **at least 160 feet** between \( t = 2 \) and \( t = 5 \).

Therefore, the time period during which the projectile is at least 160 feet above the ground is:

\[
2 \leq t \leq 5 \quad \text{seconds}.
\]

---

Would you like more details, or do you have any questions about this process?

Here are 5 related questions:
1. What is the maximum height reached by the projectile?
2. How long does it take for the projectile to reach the maximum height?
3. How long is the projectile in the air before it hits the ground?
4. What is the velocity of the projectile at \( t = 2 \) seconds?
5. How can the total flight time of the projectile be calculated?

**Tip:** When solving quadratic inequalities, always consider the physical context, such as time, to determine meaningful solutions.

A projectile is fired straight up from ground level. After t​ seconds, its height above the ground is s​ feet, where s = −16t^2 + 112t. For what time period is the projectile at least 160 feet above the​ ground?

Find the time period during which a projectile is at least 160 feet above the ground using the quadratic equation s = −16t^2 + 112t. Step-by-step solution involves solving quadratic inequalities and interpreting the results in the context of projectile motion. Ideal for students in Grades 9-12.

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