https://file.aiquickdraw.com/mathbot/file/1727266573331m556srka.jpg

We are given the function for the height \( h(x) \) of a rock, which is thrown upward from a bridge:

\[
h(x) = -x^2 + 3x + 10
\]

Where:
- \( h(x) \) is the height of the rock in feet.
- \( x \) is the time in seconds after the rock is thrown.
- The road is 10 feet below the bridge, so we need to find when \( h(x) = -10 \).

### Step 1: Set up the equation for when the rock hits the road
We know that the height of the road is 10 feet below the bridge, meaning \( h(x) = -10 \). So, we set the given function equal to -10:

\[
-x^2 + 3x + 10 = -10
\]

### Step 2: Solve the equation
First, move all terms to one side of the equation:

\[
-x^2 + 3x + 10 + 10 = 0
\]
\[
-x^2 + 3x + 20 = 0
\]

Now, multiply the equation by -1 to make the coefficient of \( x^2 \) positive:

\[
x^2 - 3x - 20 = 0
\]

### Step 3: Factor the quadratic equation
We factor the equation:

\[
(x - 5)(x + 4) = 0
\]

### Step 4: Solve for \( x \)
Set each factor equal to zero:

\[
x - 5 = 0 \quad \text{or} \quad x + 4 = 0
\]

So, \( x = 5 \) or \( x = -4 \).

Since time cannot be negative, we discard \( x = -4 \), and the solution is:

\[
x = 5 \text{ seconds}
\]

### Conclusion:
The rock will hit the road after 5 seconds.

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How would the result change if the road were at a different height?
2. What does the negative sign in front of \( x^2 \) represent in the motion of the rock?
3. Can you derive the time it takes for the rock to reach its maximum height?
4. What other methods can be used to solve quadratic equations?
5. How would the problem differ if the rock were thrown downward instead of upward?

**Tip**: Factoring is one of the simplest ways to solve quadratic equations, but if factoring is difficult, you can always use the quadratic formula!

A rock is thrown upward from the edge of a bridge and onto a road that is 10 feet below the bridge. The function h(x) = -x^2 + 3x + 10 gives the height, h, in feet, the rock travels in x seconds from the time it was thrown. When will the rock hit the road?

This problem involves solving a quadratic equation to find when a rock, thrown upward from a bridge, hits the ground 10 feet below. The quadratic equation h(x) = -x^2 + 3x + 10 represents the height of the rock. Learn how to solve for the time the rock hits the road using algebraic techniques.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation