Question

sarah and miguel are both walking towards the park at a constant rate as represented by the graph to the left.
part a: use the graph to write an equation that models the relationship between miguel’s distance from the park and time passed in minutes. define your variables.
part b: after how many minutes will sarah and miguel be equally distant from the park? describe how you arrived at your answer.

Explanation:

Step1: Define variables for Miguel

Let $d$ = distance from park (meters), $t$ = time passed (minutes).

Step2: Find Miguel's starting distance

At $t=0$, $d=300$ meters.

Step3: Calculate Miguel's walking rate

Rate = $\frac{\text{Change in }d}{\text{Change in }t} = \frac{300}{30} = 10$ meters/minute (decreasing).

Step4: Write Miguel's distance equation

$d = 300 - 10t$

Step5: Define variables for Sarah (same as above)

Let $d$ = distance from park (meters), $t$ = time passed (minutes).

Step6: Find Sarah's starting distance

At $t=0$, $d=600$ meters.

Step7: Calculate Sarah's walking rate

Rate = $\frac{\text{Change in }d}{\text{Change in }t} = \frac{600}{20} = 30$ meters/minute (decreasing).

Step8: Write Sarah's distance equation

$d = 600 - 30t$

Step9: Set distances equal for Part B

$300 - 10t = 600 - 30t$

Step10: Solve for $t$

$20t = 300$
$t = \frac{300}{20} = 15$

Answer:

Part A:

Miguel's equation: Let $d$ = distance from the park (meters), $t$ = time passed (minutes). $d = 300 - 10t$

Part B:

15 minutes. To find this, set the distance equations for Sarah ($d=600-30t$) and Miguel ($d=300-10t$) equal to each other, then solve for $t$: $300-10t=600-30t$, which simplifies to $20t=300$, so $t=15$.