Question

A van starts off 191 miles directly north from the city of Morristown. It travels due east at a speed of 25 miles per hour. After travelling 119 miles, how fast is the distance between the van and Morristown changing?

Answer 1

Distance between the van and Morristown is changing at the rate of 13.22 miles per hour.

Explanation:

From the figure attached,

Van starts from C (City of Morristown), reaches the point A 191 miles due North and then it travels with a speed of 25 miles per hour due East from A towards B.

We have to calculate the rate of change of distance BC, when the van reaches point B which is 119 miles away from A.

By Pythagoras theorem in the triangle ABC,

`BC^(2)=AB^(2)+AC^(2)`

Distance AC is constant equal to 191 mi.

By differentiating the equation with respect to time 't'

`2BC.(d(BC))/(dt)=0+2AB.(d(AB))/(dt)`

`BC.(d(BC))/(dt)=AB.(d(AB))/(dt)`

Since BC² = (119)²+ (191)²

BC = √50642 = 225.04 miles

From the differential equation,

`(225.03).(d(BC))/(dt)=119* 25` [Since
`(d(AB))/(dt)=25` miles per hour]

`(d(BC))/(dt)=13.22` miles per hour

Therefore, distance between the van and Morristown is changing at the rate of 13.22 miles per hour.