A Word Problem using Parametric Equations

by Amy Benson

Problem:

Two cars are racing 500 miles on a 2 mile oval track. The first car averages 105 mph. The second car averages 120 mph but is delayed at the start by 30 minutes due to engine problems.

Solution:

Begin by using the equation for distance to write the equation of a line for each car.

Distance = rate x time

y1 = 105 x

y2 = 120(x - 0.5)

Now, write the equations in parametric form.

x1 = t

y1 = 105 t

and

x2 = t

y2 = 120(t - .5)

 

This graph indicates that the intersection is (4, 420) which can be confirmed using the point slope equations.

 

Now for the winner: the car with the smaller t value when y = 500 wins the race. Let's trace on our graph and see who wins.

Tracing Car 1's graph:

Our best approximation for t when y = 500 for car 1 is t = 4.76.

Tracing Car 2's graph:

Our closet approximation for t when y = 500 for car 2 is t = 4.66 hours.

CAR 2 WINS!!

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