Question

Car A is driving east toward an intersection. Car B has already gone through the same intersection and is heading north. At what rate is the distance between the cars changing at the instant when car A is 40 miles from the intersection and traveling at 50 mph and car B is 30 miles from the intersection and traveling at 60 mph? Are the cars getting closer together or farther apart at this time?

Answer 1

Answer:

76 mi/h

Explanation:

$\frac{da}{dt}$ = Velocity of car A = 50 mi/h

a = Distance car A travels = 40 mi

$\frac{db}{dt}$ = Velocity of car B = 60 mi/h

b = Distance car B = 30 mi

c = Distance between A and B after 3 hours = √(a²+b²) = √(40²+90²) = 50 mi

From Pythagoras theorem

a²+b² = c²

Now, differentiating with respect to time

$2a\frac{da}{dt}+2b\frac{db}{dt}=2c\frac{dc}{dt}\\\Rightarrow a\frac{da}{dt}+b\frac{db}{dt}=c\frac{dc}{dt}\\\Rightarrow \frac{dc}{dt}=\frac{a\frac{da}{dt}+b\frac{db}{dt}}{c}\\\Rightarrow \frac{dc}{dt}=\frac{40\times 50+30\times 60}{50}\\\Rightarrow \frac{dc}{dt}=76\ mi/h$

∴ Rate at which distance between the cars is increasing is 76 mi/h

The cars are getting farther apart at this time