Two taxicabs begin at the same time at the intersection of Park Ave and 59th St. One taxicab heads uptown (North) on Park Ave at 40 mph. The other heads west on 59th St at 30 mph. Both have the unbelievable good fortune not to be stopped by traffic lights or slowed by traffic. When the taxicab on Park Ave. has gone half a mile, how fast is the distance (as the New York crow flies) between them increasing?
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One way to do this problem is superimpose a coordinate system. Set the intersection of Park Ave and 59th street to be at (0, 0), the origin. Then, after t hours, we see that the north-bound taxicab is at (0, 30t), while the west-bound taxi cab is at (-40t, 0). The distance between the two cabs is:
D = √[(30t)^2 + (-40t)^2] = 50t.
By differentiating, dD/dt = 50 for any value of t. Therefore, the distance between them is increasing at a rate of 50mph after one mile (and regardless of distance between them).
I hope this helps!
D = √[(30t)^2 + (-40t)^2] = 50t.
By differentiating, dD/dt = 50 for any value of t. Therefore, the distance between them is increasing at a rate of 50mph after one mile (and regardless of distance between them).
I hope this helps!