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?

If the meet after x hours, then

1.5 + 4.5x = 10x

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To find out how long Jacinda will take to catch up with Raj, we need to calculate the distances traveled by each person.

Raj gets a 1.5 mile head start, so his progress can be represented by the equation: Raj's distance = 1.5 + 4.5t (where t is time in hours).

Now let's calculate Jacinda's progress. We are given three points that represent her progress on a graph: (1,10), (2,20), and (3,30). This implies that for every hour, Jacinda covers 10 miles.

So Jacinda's progress can be represented by the equation: Jacinda's distance = 10t (where t is time in hours).

For Jacinda to catch up with Raj, their distances must be equal. We can set up an equation and solve for t:

1.5 + 4.5t = 10t

Subtracting 4.5t from both sides:

1.5 = 5.5t

Dividing both sides by 5.5:

t = 1.5 / 5.5

t ≈ 0.27

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

To solve this problem, we need to find out how long Jacinda needs to run to catch up with Raj.

First, let's find out how far Raj has already run. Given that Raj gets a 1.5-mile head start, we can add this to the distance he has run to find his current position:

Distance Raj has run = Distance of Raj's head start + Distance covered by Raj

Distance Raj has run = 1.5 miles + 4.5 miles/hour × Time

Now, let's determine Jacinda's position at any given time based on the graph represented by the points (1,10), (2,20), and (3,30). We can assume that Jacinda's position follows a linear relationship between time (x-axis) and distance (y-axis).

Using the points (1,10) and (2,20), we can determine the slope of the line, which represents Jacinda's rate:

Slope = Change in distance / Change in time

Slope = (20 - 10) / (2 - 1) = 10 / 1 = 10 miles/hour

Now, using the slope-intercept form of a linear equation, where y = mx + b (m is the slope and b is the y-intercept), we can determine Jacinda's position:

Distance Jacinda has run = Jacinda's rate × Time + y-intercept

Distance Jacinda has run = 10 miles/hour × Time + y-intercept

Using the point (1,10), we can determine the y-intercept:

10 = 10 miles/hour + y-intercept
y-intercept = 10 - 10 miles/hour = 0 miles

Therefore, the equation for Jacinda's position is:

Distance Jacinda has run = 10 miles/hour × Time

To find the time when Jacinda catches up with Raj, we need to equate their distances and solve for Time:

Distance Raj has run = Distance Jacinda has run

1.5 miles + 4.5 miles/hour × Time = 10 miles/hour × Time

1.5 miles = 5.5 miles/hour × Time

Time = 1.5 miles / 5.5 miles/hour = 0.2736... hours

To convert this time to minutes or other units, you can multiply it by 60 minutes/hour or the appropriate conversion factor.