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,
[tex]BC^{2}=AB^{2}+AC^{2}[/tex]
Distance AC is constant equal to 191 mi.
By differentiating the equation with respect to time 't'
[tex]2BC.\frac{d(BC)}{dt}=0+2AB.\frac{d(AB)}{dt}[/tex]
[tex]BC.\frac{d(BC)}{dt}=AB.\frac{d(AB)}{dt}[/tex]
Since BC² = (119)²+ (191)²
BC = √50642 = 225.04 miles
From the differential equation,
[tex](225.03).\frac{d(BC)}{dt}=119\times 25[/tex] [Since [tex]\frac{d(AB)}{dt}=25[/tex] miles per hour]
[tex]\frac{d(BC)}{dt}=13.22[/tex] miles per hour
Therefore, distance between the van and Morristown is changing at the rate of 13.22 miles per hour.
