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