Question

Two cars leave an intersection at the same time. One is headed south at a constant speed of $30$ miles per hour, and the other is headed west at a constant speed of$40$ miles per hour (see the figure). Build a model that expresses the distance between the cars as a function of the time t.

[Hint: At $t=0$, the cars leave the intersection.]

Short answer

The distance d between the cars as a function of the time tis $50t$.

Step-by-step solution

Step 1. Find the equations for the distance travelled by both cars.

The distance travelled by first car in time t is $30t$.

The distance travelled by the second car in time t is $40t$.

Step 2. Apply Pythagorean theorem in the following figure.

$$\begin{gathered} d\left(t\right)=\sqrt{{\left(30t\right)}^2+{\left(40t\right)}^2} \\ =\sqrt{900t^2+1600t^2} \\ =\sqrt{2500t^2} \\ =50t \end{gathered}$$

So the distance d between the cars as a function of the time tis $50t$.