# Calculus

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Two automobiles are travelling on intersecting roads. The first automobile is travelling northeast at 35 km/h. The second automobile begins 7 km north of the first, and it is travelling east at 25 km/h. Assuming that these cars will continue at this rate, will they collide? Assign vector equations to each line and assume that the first automobile begins at the origin.

• Calculus -

Let (0,0) be where the first car started. Then its vector will be

u = 35/√2 t i + 35/√2 t j
= 24.75ti + 24.75tj

The second car starts at (0,7), so its vector is

v = 25ti + 7j

Is there a value for t where the two are equal? Let's see.

It will take car1 7/24.78=.2828 hr to travel 7km east and 7km north
It will take car2 7/25=.2800 hr to travel 7km east, staying at 7km north.

So, since .0028 hrs is about 10 seconds, there probably won't be a crash.

• Calculus - PS -

Hmm. I see a typo. Car 1 travels at 24.7487, not 24.78km/hr

So, it takes 7/(35/√2) = √2/5 = 0.2828 hrs. Same answer.

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