To determine the time of splashdown and the maximum height of the rocket, we start with the height function given by:
\[ h(t) = -4.9t^2 + 49t + 283 \]
Part 1: Time of Splashdown
The splashdown occurs when the height \( h(t) \) is equal to 0. We need to solve the equation:
\[ -4.9t^2 + 49t + 283 = 0 \]
We can use the quadratic formula to find the values of \( t \):
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = -4.9 \), \( b = 49 \), and \( c = 283 \). Now we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = 49^2 - 4 \cdot (-4.9) \cdot 283 \]
Calculating \( 49^2 \):
\[ 49^2 = 2401 \]
Now, calculate \( -4 \cdot (-4.9) \cdot 283 \):
\[ -4 \cdot (-4.9) \cdot 283 = 19.6 \cdot 283 = 5554.8 \]
So,
\[ b^2 - 4ac = 2401 + 5554.8 = 7955.8 \]
Now substituting back into the quadratic formula:
\[ t = \frac{-49 \pm \sqrt{7955.8}}{2 \cdot -4.9} \]
Calculating \( \sqrt{7955.8} \):
\[ \sqrt{7955.8} \approx 89.18 \]
Now substituting:
\[ t = \frac{-49 \pm 89.18}{-9.8} \]
Calculating the two possible solutions for \( t \):
- \( t_1 = \frac{-49 + 89.18}{-9.8} \)
- \( t_2 = \frac{-49 - 89.18}{-9.8} \)
Calculating \( t_1 \):
\[ t_1 = \frac{40.18}{-9.8} \approx -4.1 \quad (\text{Not a valid solution as time cannot be negative}) \]
Calculating \( t_2 \):
\[ t_2 = \frac{-138.18}{-9.8} \approx 14.1 \]
Thus, the splashdown occurs at approximately:
\[ \text{The rocket splashes down after } 14.1 \text{ seconds.} \]
Part 2: Maximum Height
The maximum height of a parabolic function \( h(t) = at^2 + bt + c \) occurs at the vertex. The time at which the maximum height occurs can be found using:
\[ t = -\frac{b}{2a} \]
Substituting \( a = -4.9 \) and \( b = 49 \):
\[ t = -\frac{49}{2 \cdot -4.9} = \frac{49}{9.8} \approx 5 \]
Now to find the maximum height, we substitute \( t = 5 \) back into the height function:
\[ h(5) = -4.9(5^2) + 49(5) + 283 \]
Calculating \( 5^2 = 25 \):
\[ h(5) = -4.9(25) + 245 + 283 \]
Now calculating:
\[ h(5) = -122.5 + 245 + 283 = 405.5 \]
So, the maximum height the rocket reaches is:
\[ \text{The rocket peaks at } 405.5 \text{ meters above sea level.} \]
Summary of Answers
- The rocket splashes down after 14.1 seconds.
- The rocket peaks at 405.5 meters above sea level.