Duffer McGee stood on a hill and used a nine iron to hit a golf ball that reached a maximum height of 126 feet and stayed in the air for 5.2 seconds before it touched the ground.

Mars has a gravity of approximately 12 feet per second squared compared to Earth's 32 feet per second squared. NASA did a simulation to try to determine how high the golf ball would fly and how long it would stay in the air on Mars if it was hit at the same height, angle, and velocity as Duffer's. The data below represent the results of that simulation:

\begin{tabular}{|c|r|r|r|r|r|}
\hline
[tex]$t$[/tex] & 1 & 2 & 3 & 4 & 5 \\
\hline
[tex]$H(t)$[/tex] & 106 & 164 & 210 & 244 & 266 \\
\hline
\end{tabular}

1. Use the Quadratic Regression feature of your calculator to generate a mathematical model for this situation. Write the function below. Round each coefficient to the nearest whole number.

[tex]\[
H(t) = \boxed{}
\][/tex]

2. Based on your model, how high is the hill from which the golf ball was hit?

The golf ball was hit from a hill [tex]$\boxed{}$[/tex] feet high.

3. Use your model to estimate how long the golf ball will take to reach its maximum height and what its maximum height will be. Round your answers to two decimal places.

The golf ball will reach a maximum height of [tex]$\boxed{}$[/tex] feet after [tex]$\boxed{}$[/tex] seconds.

4. Use your model to determine how long it will take for the golf ball to hit the surface of Mars. Round your answer to two decimal places.

The golf ball will reach the surface of Mars after [tex]$\boxed{}$[/tex] seconds.



Answer :

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.