Example  1
  1. Adrienne and Eric are at the top of a slope, and plan to have a race. Adrienne will coast her bicycle down the slope and Eric will run. Adrienne has agreed to give Eric a \(5\) m head start. The distance, \(d\), in metres, at time \(t\) seconds, can be modelled using the following equations:

    Adrienne: \(d = 1.25t^2 + 0.2t\)

    Eric: \(d = 7.5t + 5\)

    1. After how much time will Adrienne and Eric have travelled the same distance? Round the answer to the nearest hundredth.

      Substitute the second equation for \(d\), and solve for \(t\).

      \(\begin{align}
       d &= 1.25t^2 + 0.2t \\
       7.5t + 5 &= 1.25t^2 + 0.2t \\
       0 &= 1.25t^2 - 7.3t - 5 \\
       \end{align}\)

      \[\begin{align}
       t &= \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{{2a}} \\
        &= \frac{{ - \left( { - 7.3} \right) \pm \sqrt {\left( { - 7.3} \right)^2 - 4\left( {1.25} \right)\left( { - 5} \right)} }}{{2\left( {1.25} \right)}} \\
        &= \frac{{7.3 \pm \sqrt {78.29} }}{{2.5}} \\
       \end{align}\]

      \(t \doteq -0.62\) or \(t \doteq 6.46\)

      The negative solution is discarded because a negative time does not make sense in the context of this situation.

      Adrienne and Eric will have travelled the same distance approximately \(6.46\) s after the race begins.
    2. If the slope is \(60\) m long, who will win the race?

      Substitute \(60\) for \(d\) in each equation, and solve for \(t\).

      \(\begin{align}
       60 &= 7.5t + 5 \\
       55 &= 7.5t \\
       7.33 &\doteq t
       \end{align}\)
      \[\begin{align}
       60 &= 1.25t^2 + 0.2t \\
       0 &= 1.25t^2 + 0.2t - 60 \\
       t &= \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{{2a}} \\
        &= \frac{{ - \left( {0.2} \right) \pm \sqrt {\left( {0.2} \right)^2 - 4\left( {1.25} \right)\left( { - 60} \right)} }}{{2\left( {1.25} \right)}} \\
        &= \frac{{ - 0.2 \pm \sqrt {300.04} }}{{2.5}} \\
       t &\doteq 6.85
       \end{align}\]

      Adrienne will reach the bottom of the slope in approximately \(6.85\) s, while Eric will take approximately \(7.33\) s to travel the same distance, so Adrienne will win the race.