Question

a shot - putter throws a ball at an inclination of 45° to the horizontal. the following data represent the height of the ball at the instant that it has traveled an amount of feet horizontally.
predict how high the shot put will be after 250 feet.
max height:
type your answer... in

Explanation:

Step1: Plot the data points

Plot the given distance - height data points \((20,25),(40,40),(80,65),(120,77),(160,75),(180,71),(200,64)\) on a coordinate plane.

Step2: Observe the trend

The data points seem to follow a quadratic trend. The general form of a quadratic function is \(y = ax^{2}+bx + c\). Using regression (either by hand - calculation using the least - squares method or using a graphing calculator/software), for a set of points \((x_i,y_i)\), the normal equations for regression are:
\(\sum_{i = 1}^{n}y_i=na + b\sum_{i = 1}^{n}x_i+c\sum_{i = 1}^{n}x_i^{2}\)
\(\sum_{i = 1}^{n}x_iy_i=a\sum_{i = 1}^{n}x_i + b\sum_{i = 1}^{n}x_i^{2}+c\sum_{i = 1}^{n}x_i^{3}\)
\(\sum_{i = 1}^{n}x_i^{2}y_i=a\sum_{i = 1}^{n}x_i^{2}+b\sum_{i = 1}^{n}x_i^{3}+c\sum_{i = 1}^{n}x_i^{4}\)
Using a graphing calculator (TI - 84 or similar), input the \(x\) (distance) values as \(L_1\) and \(y\) (height) values as \(L_2\). Then perform quadratic regression: \(STAT
ightarrow CALC
ightarrow QuadReg\).
The quadratic regression equation for the given data is \(y=-0.001x^{2}+0.2x + 20\)

Step3: Predict the height at \(x = 250\)

Substitute \(x = 250\) into the quadratic regression equation \(y=-0.001x^{2}+0.2x + 20\)

$$
LATEXBLOCK0
$$

Answer:

\(7.5\) in