Question: An object is launched from a platform.
Its height (in meters), $x$ seconds after the launch, is modeled by
$$h(x) = -5(x + 1)(x - 9)$$
How many seconds after launch will the object hit the ground?
seconds
1. **State the problem:** We want to find the time $x$ when the object hits the ground, which means when the height $h(x)$ is zero.
2. **Write the equation:** The height is given by $$h(x) = -5(x + 1)(x - 9).$$ We need to solve for $x$ when $$h(x) = 0.$$
3. **Set the height to zero:**
$$0 = -5(x + 1)(x - 9).$$
4. **Divide both sides by $-5$ to simplify:**
$$\cancel{-5} \cdot (x + 1)(x - 9) = \cancel{-5} \cdot 0 \implies (x + 1)(x - 9) = 0.$$
5. **Use the zero product property:** For the product to be zero, either
$$x + 1 = 0 \quad \text{or} \quad x - 9 = 0.$$
6. **Solve each equation:**
- If $x + 1 = 0$, then $$x = -1.$$
- If $x - 9 = 0$, then $$x = 9.$$
7. **Interpret the solutions:** Since $x$ represents time after launch, negative time $x = -1$ seconds is not physically meaningful.
8. **Final answer:** The object hits the ground $$9$$ seconds after launch.