Question

Two cars start moving from the same point. One travels south at 24 mi/h and the other travels west at 18 mi/h. At what rate is the distance between the cars increasing three hours later?

Answer 1

in 1 hours 6 in 3 its 18

Answer 2

30 mi/h

Explanation:

`(da)/(dt)` = Velocity of car A = 24 mi/h

a = Distance car A travels in 3 hours = 24×3 = 72 mi

`(db)/(dt)` = Velocity of car B = 18 mi/h

b = Distance car B travels in 3 hours = 18×3 = 54 mi

c = Distance between A and B after 3 hours = √(a²+b²) = √(72²+54²) = 90 mi

From pythogoras theorem

a²+b² = c²

Now, differenciating with respect to time

`2a(da)/(dt)+2b(db)/(dt)=2c(dc)/(dt)\\\Rightarrow a(da)/(dt)+b(db)/(dt)=c(dc)/(dt)\\\Rightarrow (dc)/(dt)=(a(da)/(dt)+b(db)/(dt))/(c)\\\Rightarrow (dc)/(dt)=(18* 54+24* 72)/(90)\\\Rightarrow (dc)/(dt)=30\ mi/h`

∴ Rate at which distance between the cars is increasing three hours later is 30 mi/h