Question

question #20
the graph below shows the relationship between the times and distances danielle and dorian ran along the same trail. danielle ran 1 mile every 12 minutes. dorian started running once danielle had completed 1 mile, and then he ran 1 mile every 6 minutes.
danielle’s and dorian’s run
how far had danielle run when dorian caught up to her?
a 1 mile
b 2 miles
c 12 miles
d 24 miles

Explanation:

Step1: Define Danielle's distance function

Danielle's speed is $\frac{1}{12}$ miles per minute. Let $t$ = total time (minutes). Her distance $d_D$ is:
$d_D = \frac{1}{12}t$

Step2: Define Dorian's distance function

Dorian starts at $t=12$ minutes (when Danielle finishes 1 mile). His speed is $\frac{1}{6}$ miles per minute. His time running is $t-12$, so his distance $d_o$ is:
$d_o = \frac{1}{6}(t-12)$

Step3: Set distances equal (catch-up)

When Dorian catches up, $d_D = d_o$:
$\frac{1}{12}t = \frac{1}{6}(t-12)$

Step4: Solve for total time $t$

Multiply both sides by 12 to eliminate denominators:
$t = 2(t-12)$
$t = 2t - 24$
$t = 24$ minutes

Step5: Find Danielle's distance at $t=24$

Substitute $t=24$ into Danielle's function:
$d_D = \frac{1}{12} \times 24 = 2$

Answer:

B. 2 miles