Question

Beginning 156 miles directly east of the city of Uniontown, a truck travels due south. If the truck is travelling at a speed of 31 miles per hour, determine the rate of change of the distance between Uniontown and the truck when the truck has been travelling for 81 miles. (Do not include units in your answer, and round to the nearest tenth.)

Answer 1

Answer:

14.3

Explanation:

The distance s as a function of time can be written as:

$s(t) = \sqrt{156^{2} + (31t)^2}$

The rate of change is the derivative of d with respect to time:

$\frac{ds}{dt} =\frac{961t}{\sqrt{156^{2}+(31t)^{2}}}$

The time t when the track has been traveling for 81 miles is given by:

$81 = 31t\\ t = \frac{81}{31}$

Using t in the previous equation gives:

$\frac{ds}{dt}(\frac{81}{31} ) =\frac{2511}{\sqrt{156^{2}+81^{2}}}=14.3$