1. Car A is traveling north on road $1$ at $35$ miles per hour. Car B is traveling east on road $2$ at $65$ miles per hour. How fast is the distance between them shrinking when car A is one mile south and car B is one mile west of the intersection of roads $1$ and $2$?
  2. Suppose at t seconds that Sally is travelling $t^3 m/s$ and Fred is traveling $t^2 m/s$. How much farther than Fred has Sally traveled when $t=3$ seconds?

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For the first question, write the distances of cars A and B from the intersection as functions of time, say $A(t)$ and $B(t)$. You're given that $$A'(t) = -35 \,\text{mph}$$ $$B\,'(t) = -65 \,\text{mph}$$ Then the distance between the cars, by the Pythagorean Theorem, equal to $$D(t) = \sqrt{A(t)^2 + B(t)^2}$$

The question asks for the value of $D\,'(t)$, when $A(t) = 1$ and $B(t) = 1$. For convenience, set $A(0) = 1$ and $B(0)=1$.

Then we want $D\,'(0)$.

For the second question, I'll assume that you start at $t=0$ seconds. Use the fact that the distance covered is equal to the integral of the velocity with respect to time.

In this case, the bounds on your integrals (for both Fred and Sally) will be $t=0$ and $t=3$.

Good luck!

  • $\begingroup$ Thank you for your help.For the first one, why t is 3? $\endgroup$ – user113351 Dec 4 '13 at 8:28
  • $\begingroup$ Oops! I mixed up the parts. I will edit it so that it reads correctly. I also fixed the signs of the derivatives of $A(t)$ and $B(t)$. Since the distances of both cars from the intersection are decreasing, the derivatives of these distances should be negative. $\endgroup$ – Zubin Mukerjee Dec 4 '13 at 8:29

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