Question

grade 8 unit 4
georgia’s k-12 mathematics standards
name: ______________________
date: ______________________
driving home
john is 150 miles from atlanta and is driving an average of 30 mph. jacob is 200 miles from atlanta is driving an average of 40 mph.
a. write a linear function that models john’s journey from where he is to atlanta.
b. write a linear function that models jacob’s journey from where he is to atlanta.
c. use your linear functions to determine when john and jacob will be the same distance away from atlanta.
d. create a graph that represents both linear functions and proves your solution for part c.
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Explanation:

Step1: Define variables

Let $t$ = time in hours, $d_J(t)$ = John's distance from Atlanta, $d_Ja(t)$ = Jacob's distance from Atlanta.

Step2: Model John's journey

John starts 150 miles from Atlanta, drives toward Atlanta at 30 mph, so distance decreases over time.
$d_J(t) = 150 - 30t$

Step3: Model Jacob's journey

Jacob starts 200 miles from Atlanta, drives toward Atlanta at 80 mph, so distance decreases over time.
$d_Ja(t) = 200 - 80t$

Step4: Find equal distance time

Set the two functions equal and solve for $t$:
$150 - 30t = 200 - 80t$
Add $80t$ to both sides:
$150 + 50t = 200$
Subtract 150 from both sides:
$50t = 50$
Divide by 50:
$t = 1$

Step5: Graph setup (key points)

• For $d_J(t)=150-30t$:
• When $t=0$, $d=150$; when $d=0$, $t=5$ (x-intercept: $(5,0)$, y-intercept: $(0,150)$)
• For $d_Ja(t)=200-80t$:
• When $t=0$, $d=200$; when $d=0$, $t=2.5$ (x-intercept: $(2.5,0)$, y-intercept: $(0,200)$)
• The lines intersect at $(1, 120)$, proving the solution to part c.

Answer:

a. $d_J(t) = 150 - 30t$
b. $d_Ja(t) = 200 - 80t$
c. After 1 hour, John and Jacob will be the same distance from Atlanta.
d. A graph with:

• Line 1 (John): connects $(0, 150)$ and $(5, 0)$
• Line 2 (Jacob): connects $(0, 200)$ and $(2.5, 0)$
• Intersection point at $(1, 120)$, confirming the 1-hour time when their distance from Atlanta is equal (120 miles).