Question

Car A is traveling north on a straight highway and car B is traveling west on a different straight highway. Each car is approaching the intersection of these highways. At a certain moment, car A is 0.4 km from the intersection and traveling at 75 km/h while car B is 0.3 km from the intersection and traveling at 70 km/h. How fast is the distance between the cars changing at that moment? km/h

Answer 1

102 km/h

Step-by-step explanation:

x = distance of car A at any time from the intersection

x₀ = distance of car A at some time = 0.4 km

`v_(A)` = Speed of car A = 75 km/h

y = distance of car B at any time from the intersection

y₀ = distance of car B at some time = 0.3 km

`v_(B)` = Speed of car B = 70 km/h

d = distance between the two cars at any time

d₀ = distance between the two cars at some time

v = rate of change of distance between the cars

Using Pythagorean theorem

d²₀ = x₀² + y₀²

d²₀ = 0.4² + 0.3²

d₀ = 0.5 m

Distance between the two cars at any time is given using Pythagorean theorem as

d² = x² + y²

Taking derivative both side relative to "t"

`2d \left ( (dd)/(dt) \right ) = 2x ( (dx)/(dt) \right ) + 2y ( (dy)/(dt) \right )`

`d_(o) v = x_(o) ( (dx)/(dt) \right ) + y_(o) ( (dy)/(dt) \right )`

(0.5) v = (0.4)
`v_(A)` + (0.3)
`v_(B)`

(0.5) v = (0.4) (75) + (0.3) (70)

v = 102 km/h