Question

multiple choice 1 point a pumpkin tossing contest is held each year in morton, illinois, where people compete to see whose catapult will send pumpkins the farthest. one catapult launches pumpkins from 25 feet above the ground at a speed of 125 feet per second. the table shows the horizontal distances (in feet) the pumpkins travel when launched at different angles. angle (degrees) 20 30 40 50 60 70 distance (feet) 372 462 509 501 437 323 which function best models the data? y = -0.261x² + 22.591x - 23.029 y = -0.261x² + 22.591x y = 22.591x² + 23.029 y = -0.261x² + 22.591x + 23.029

Explanation:

Step1: Test x=20 in each function

For $y=-0.261x^2 + 22.591x - 23.029$:

$$\begin{align*} y&=-0.261(20)^2 + 22.591(20) - 23.029\\ &=-0.261(400)+451.82-23.029\\ &=-104.4+451.82-23.029=324.391 \end{align*}$$

For $y=-0.261x^2 + 22.591x$:

$$\begin{align*} y&=-0.261(20)^2 + 22.591(20)\\ &=-104.4+451.82=347.42 \end{align*}$$

For $y=22.591x^2 + 23.029$:

$$\begin{align*} y&=22.591(20)^2 +23.029\\ &=22.591(400)+23.029=9036.4+23.029=9059.429 \end{align*}$$

For $y=-0.261x^2 + 22.591x +23.029$:

$$\begin{align*} y&=-0.261(20)^2 +22.591(20)+23.029\\ &=-104.4+451.82+23.029=370.449 \end{align*}$$

Step2: Match to table value (372)

The result $370.449$ is closest to 372, corresponding to $y=-0.261x^2 + 22.591x +23.029$. Verify with x=30:

$$\begin{align*} y&=-0.261(30)^2 +22.591(30)+23.029\\ &=-0.261(900)+677.73+23.029\\ &=-234.9+677.73+23.029=465.859 \end{align*}$$

This is close to the table value 462, confirming the model.

Answer:

D. $y = -0.261x^2 + 22.591x + 23.029$