The problem involves solving for $t$ when the ball's height $h$ equals 4 meters. The given equation for the height of the ball is:

$h = 1 + 9t - 5t^2$

Step 1: Set $h = 4$

Substitute $h = 4$ into the equation: $4 = 1 + 9t - 5t^2$

Simplify: $0 = -5t^2 + 9t - 3$

Rearrange: $5t^2 - 9t + 3 = 0$

Step 2: Solve the quadratic equation

The quadratic equation is: $5t^2 - 9t + 3 = 0$

We use the quadratic formula: $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 5$, $b = -9$, and $c = 3$.

Step 3: Calculate discriminant

$\text{Discriminant} = b^2 - 4ac = (-9)^2 - 4(5)(3)$ $\text{Discriminant} = 81 - 60 = 21$

Step 4: Find $t$

$t = \frac{-(-9) \pm \sqrt{21}}{2(5)}$ $t = \frac{9 \pm \sqrt{21}}{10}$

Now calculate the two solutions: $t_1 = \frac{9 + \sqrt{21}}{10}, \quad t_2 = \frac{9 - \sqrt{21}}{10}$

1. Compute $\sqrt{21}$: $\sqrt{21} \approx 4.5826$

2. Solve for $t_1$: $t_1 = \frac{9 + 4.5826}{10} = \frac{13.5826}{10} \approx 1.36$

3. Solve for $t_2$: $t_2 = \frac{9 - 4.5826}{10} = \frac{4.4174}{10} \approx 0.44$

Final Answer:

$t = 0.44 \, \text{or} \, t = 1.36$

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Related Questions:

1. What is the meaning of the discriminant in quadratic equations?
2. How can you verify the solutions to a quadratic equation?
3. Why does the equation have two solutions for $t$?
4. What happens to the motion of the ball after it reaches its peak height?
5. How would air resistance affect this problem?

Tip:

Always double-check your solutions by substituting them back into the original equation!