Two cars are approaching an intersection. One is 3 miles south of the intersection and is moving at a constant speed of 10 miles per hour. At the same​ time, the other car is 22 miles east of the intersection and is moving at a constant speed of 40 miles per hour.

Question

(a) Express the distance d between the cars as a function of time t. ​ (Hint: At t = 0, the cars are 3 milesmiles south and 2 milesmiles east of the​ intersection, respectively.)

Answer 1

To find the distance between the two cars as a function of time, we can use the Pythagorean theorem. Let $d$ represent the distance between the two cars, $x$ represent the distance the first car has traveled, and $y$ represent the distance the second car has traveled.

At time $t$, the first car has traveled $x=10t$ miles, and the second car has traveled $y=40t$ miles. Using the Pythagorean theorem, we have

$$d^2=(22-40t{)}^2+(3-10t{)}^2$$

Expanding this equation gives

$$d^2=484-880t+1600t^2+9-60t+100t^2$$

Simplifying further,

$$d^2=1709-940t+1700t^2$$

Therefore, the distance between the two cars as a function of time is

$$d(t)=\sqrt{1709-940t+1700t^2}$$