Raj gets a 1.5 mile head start and runs at a rate of 4.5 miles per hour. Jacinda's progress is represented by a graph that goes through the points (1,10), (2,20), and (3,30). How long will Jacinda need to run to catch up with Raj?

Jacinda appears to be running 10 minute miles ... 6 mph

after 1 hr, Raj has gone 6 mi
... that's when Jacinda catches him

To find out how long Jacinda will need to run to catch up with Raj, we need to compare their distances at the same time. Let's break down the information we have:

- Raj gets a head start of 1.5 miles.
- Raj runs at a rate of 4.5 miles per hour.
- Jacinda's progress is represented by the points (1,10), (2,20), and (3,30), which seem to indicate that she is running at a constant rate.

To determine when Jacinda will catch up with Raj, we need to find the point on Jacinda's graph where her distance equals Raj's distance. Since Raj is running at a constant rate, his distance can be calculated using the formula: distance = rate × time.

Let's calculate Raj's distance when Jacinda has run for time 't':

Raj's distance = 1.5 miles (head start) + 4.5 miles per hour (rate) × t (time)

Now, let's analyze Jacinda's progress:

From the given points (1,10), (2,20), and (3,30), we can observe that Jacinda's distance is increasing linearly. This means Jacinda is running at a constant rate.

The rate at which Jacinda is running can be calculated from the change in distance divided by the change in time between two consecutive points.

Rate = (Change in distance) / (Change in time)

Using the first two points, we can calculate Jacinda's rate:

Rate = (20 - 10) / (2 - 1) = 10 miles per hour

Now that we have the rates and distance for both Raj and Jacinda, we can set up an equation to find the time at which Jacinda catches up with Raj:

1.5 + 4.5t = 10t

Simplifying the equation:

1.5 = (10 - 4.5)t

1.5 = 5.5t

t = 1.5 / 5.5 ≈ 0.27 hours

Therefore, Jacinda will need to run for approximately 0.27 hours (or 16 minutes) to catch up with Raj.