We are given the time [tex]\( t \)[/tex] and the height [tex]\( H(t) \)[/tex] of a golf ball hit on Mars and asked to generate a quadratic model for the situation.
### Step-by-Step Solution
1. Generate the Quadratic Model:
To find the quadratic model [tex]\( H(t) = at^2 + bt + c \)[/tex], we need to perform quadratic regression on the given data points:
[tex]\[
\begin{array}{|c|r|r|r|r|r|}
\hline
t & 1 & 2 & 3 & 4 & 5 \\
\hline
H(t) & 106 & 164 & 210 & 244 & 266 \\
\hline
\end{array}
\][/tex]
The quadratic regression yields the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
a \approx -6, \quad b \approx 76, \quad c \approx 36
\][/tex]
Thus, the quadratic model is:
[tex]\[
H(t) = -6t^2 + 76t + 36
\][/tex]
2. Determine the Height of the Hill:
To find the height of the hill, we need to evaluate the function [tex]\( H(t) \)[/tex] at [tex]\( t = 0 \)[/tex]:
[tex]\[
H(0) = -6(0)^2 + 76(0) + 36 = 36
\][/tex]
The golf ball was hit from a hill 36 feet high.
3. Time and Height of Maximum Elevation:
The time at which the golf ball reaches its maximum height (vertex of the parabola) is given by:
[tex]\[
t_{\text{max}} = -\frac{b}{2a} = -\frac{76}{2(-6)} = 6.33 \text{ seconds}
\][/tex]
To find the maximum height, we evaluate [tex]\( H(t) \)[/tex] at [tex]\( t = t_{\text{max}} \)[/tex]:
[tex]\[
H(6.33) = -6(6.33)^2 + 76(6.33) + 36 \approx 276.67 \text{ feet}
\][/tex]
The golf ball will reach a maximum height of 276.67 feet after 6.33 seconds.
4. Total Time to Hit the Surface of Mars:
To find when the ball hits the ground, we solve for [tex]\( t \)[/tex] when [tex]\( H(t) = 0 \)[/tex]:
[tex]\[
-6t^2 + 76t + 36 = 0
\][/tex]
Solving this quadratic equation gives two roots, and we consider the positive root which represents the time when the ball hits the ground:
[tex]\[
t_{\text{surface}} \approx 13.12 \text{ seconds}
\][/tex]
The golf ball will reach the surface of Mars after 13.12 seconds.
### Final Answers:
- The quadratic model is [tex]\( H(t) = -6t^2 + 76t + 36 \)[/tex].
- The golf ball was hit from a hill 36 feet high.
- The golf ball will reach a maximum height of 276.67 feet after 6.33 seconds.
- The golf ball will reach the surface of Mars after 13.12 seconds.
