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

Answer:

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

Step-by-step 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'

$2BC.\frac{d(BC)}{dt}=0+2AB.\frac{d(AB)}{dt}$

$BC.\frac{d(BC)}{dt}=AB.\frac{d(AB)}{dt}$

Since BC² = (119)²+ (191)²

BC = √50642 = 225.04 miles

From the differential equation,

$(225.03).\frac{d(BC)}{dt}=119\times 25$ [Since $\frac{d(AB)}{dt}=25$ miles per hour]

$\frac{d(BC)}{dt}=13.22$ miles per hour

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