Question

the table below shows the altitude of a plane once it begins its descent to the runway.

time (minutes) 0 1 2 3 4 5
altitude (feet) 28,500 26,378 24,105 21,774 19,452 17,991

a) using linear regression, write an equation to model the data in terms of x and y. round all numbers to the nearest hundredth. do not use any spaces in your equation.
y=

b) find the height of the plane after 8 minutes.

c) how long will it take for the plane to land?

Explanation:

Step1: Recall linear - regression formula

The linear - regression equation is of the form $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. We can use a calculator or statistical software to find $m$ and $b$ for the given data points $(x,y)$ where $x$ is time (in minutes) and $y$ is altitude (in feet). Let $x_1 = 0,y_1=28500,x_2 = 1,y_2 = 26378,\cdots,x_5=5,y_5 = 17991$.
The formula for the slope $m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}$ and $b=\overline{y}-m\overline{x}$, where $n$ is the number of data points, $\overline{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$ and $\overline{y}=\frac{\sum_{i = 1}^{n}y_i}{n}$.
Using a calculator or software (e.g., Excel, Python's numpy and scipy.stats), we find that $m\approx - 2119.42$ and $b\approx28500$. So the linear - regression equation is $y=-2119.42x + 28500$.

Step2: Find altitude at $x = 8$

Substitute $x = 8$ into the equation $y=-2119.42x + 28500$.
$y=-2119.42\times8 + 28500$.
$y=-16955.36+28500$.
$y = 11544.64\approx11544.64$ feet.

Step3: Find time to land

The plane lands when $y = 0$. Set $y = 0$ in the equation $y=-2119.42x + 28500$.
$0=-2119.42x + 28500$.
$2119.42x=28500$.
$x=\frac{28500}{2119.42}\approx13.44$ minutes.

Answer:

a) $y=-2119.42x + 28500$
b) $11544.64$
c) $13.44$