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37. Construct Arguments Luis and Raul are
riding their bicycles to the beach from their
respective homes. Luis proposes that they leave
their respective homes at the same time and
plan to arrive at the beach at the same time.
The diagram shows Luis's position at two points
during his ride to the beach.
Write an equation in slope-intercept form
to represent Luis's ride from his house to the
beach. If Raul lives 5 miles closer to the beach
than Luis, at what speed must Raul ride for the
plan to work?

37 Construct Arguments Luis and Raul are riding their bicycles to the beach from their respective homes Luis proposes that they leave their respective homes at class=

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Using the slope - intercept relation, the required equation which models the scenario and Raul's speed are ;

  • y = - 7.5x + 15
  • 4 miles per hour

Time difference, Δt = 1.2 hours - 0.5 hours = 0.7 hours

Change in distance, Δd = 11.25 - 6 = 5.25 miles

Assuming a constant speed :

  • Speed = (Δd ÷ Δt) = (5.25 ÷ 0.7) = 7.5 mi/hr

Using the general form :

  • y = bx + c

At, x = 1.2 hours ;

Miles left, y = 6 miles

End point decreases by 7.5 mi/hr (-7.5 mi/hr)

Inputting the data into the equation :

6 = - 7.5(1.2) + c

6 = - 9 + c

c = 6 + 9 = 15 miles

The expression in slope intercept form becomes ;

  • y = -7.5x + 15

If Raul lives 5 miles closer to the beach ;

Time it will take Luis to get to the beach :

  • Time taken = (15 ÷ 7.5) = 2.5 hours

Distance Raul has to cover = 15 - 5 = 10 miles

To reach the beach after 2.5 hours ;

  • Speed required = (10 ÷ 2.5) = 4 mi/hr

Therefore, Raul has to ride at 4 miles per hour for the plan to work.

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