Question

practice writing an equation for a linear function from a verbal description
study the example showing how to write an equation for a linear function from a verbal description. then solve problems 1–4.
example
dolores is making a music video using a drone. she sets the drone on a platform 1 meter above the ground. then she uses the controls to make it rise at a constant rate. the drone reaches a height of 16 meters in 5 seconds. what is the equation for the drone’s height, y, as a function of time, x?
at 0 seconds, the drone is 1 meter above the ground.
at 5 seconds, the drone is 16 meters above the ground.
rate of change: \\(\frac{16 - 1}{5 - 0} = \frac{15}{5} = 3\\) initial value: 1
use the equation for a linear function, \\(y = mx + b\\).
\\(y = 3x + 1\\)
1 the drone in the example hovers at 16 meters for a few minutes before being lowered at a constant rate. it reaches the ground after 6 seconds.
a. why can the drone’s descent be modeled by a linear function?
b. the linear model of the drone’s descent gives its height as a function of time. is the rate of change positive or negative? explain.
c. what equation models the drone’s descent as time increases?
show your work

Explanation:

Part (a)

A linear function is appropriate when a quantity changes at a constant rate. The drone's descent is at a constant rate (given as "lowered at a constant rate"), so the relationship between height and time has a constant rate of change, fitting the linear function model (since linear functions have constant slopes/rate of change).

The rate of change (slope) in a linear function \( y = mx + b \) represents how \( y \) (height) changes with \( x \) (time). As the drone descends, its height decreases over time. So when time (\( x \)) increases, height (\( y \)) decreases, meaning the rate of change (\( m \)) is negative (since a negative slope indicates a decrease in \( y \) as \( x \) increases).

Step1: Identify known points

The drone starts descending from 16 meters (initial height when descent begins, so at \( x = 0 \) seconds, \( y = 16 \)) and reaches the ground (\( y = 0 \)) at \( x = 6 \) seconds. So we have two points: \( (0, 16) \) and \( (6, 0) \).

Step2: Calculate rate of change (slope \( m \))

The formula for slope \( m \) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Substituting \( (x_1, y_1)=(0, 16) \) and \( (x_2, y_2)=(6, 0) \):
\( m=\frac{0 - 16}{6 - 0}=\frac{-16}{6}=-\frac{8}{3} \)

Step3: Determine the linear equation

Using the slope - intercept form \( y = mx + b \). The \( y \)-intercept \( b \) is the value of \( y \) when \( x = 0 \), which is 16 (from the point \( (0, 16) \)). Substituting \( m = -\frac{8}{3} \) and \( b = 16 \) into \( y = mx + b \):
\( y=-\frac{8}{3}x + 16 \)

Answer:

The drone’s descent can be modeled by a linear function because it is lowered at a constant rate. A linear function requires a constant rate of change (slope), and the drone’s height changes by a fixed amount per unit time during descent.

Part (b)