Chapter 6, Problem 9CA

To determine

To create: The scatter plot of the data, use technology to find a model for the data and then find the angles at which a launched pumpkin travels 500 ft.

Answer to Problem 9CA

  ${37}^{\circ }{\text{ and 50}}^{\circ }$

Explanation of Solution

Given: The table shows the horizontal distances y (in feet) a pumpkin travels when launched at different angle x (in degree)

  $\begin{array}{ccccccc}\text{Angle }\left(\text{Degree}\right),\mathrm{x}& 20& 30& 40& 50& 60& 70\\ \text{Distance}\left(\text{feet}\right),\mathrm{y}& 372& 462& 509& 501& 437& 323\end{array}$

Plot the scatter plot on graph.

  

According scatter plot, the data showing quadratic relationship between x and y .

Using regression technique to find the equation of quadratic relation.

  $\mathrm{y}=-0.2614{\mathrm{x}}^2+22.591\mathrm{x}+23.029$

Put $\mathrm{y}=500$

  $\begin{array}{c}500=-0.2614{\mathrm{x}}^2+22.591\mathrm{x}+23.029\\ 0.2614{\mathrm{x}}^2-22.591\mathrm{x}+476.971=0\\ \mathrm{x}=\frac{22.591\pm \sqrt{{22.591}^2-4\left(0.2614\right)\left(476.971\right)}}{2\left(0.2614\right)}\\ \mathrm{x}=36.69,49.74\\ \mathrm{x}\approx 37,50\end{array}$

Hence, the angle at which a launched pumpkin travels 500 feet are ${37}^{\circ }{\text{ and 50}}^{\circ }$ .