Rate distance and time equations

Question

Two buses leave a station at the same time and travel in opposite directions. One bus travels 15 mi h faster than the other. If the two buses are 315 miles aparte after 3 hours, what is the rate of each bus?

Answer 1

Hi Terrence,

Let's call the busses bus 1 and bus 2. Let's also make bus 1 the faster bus. Bus 1's speed is:

s₁ = 15 + s₂

The distance between the buses as a function of time (t) is:

d(t) = s₁t + s₂t

We can express this equation strictly in terms of s₂ and t because we already came up with a relation between s₁ and s₂ (s₁ = 15 + s₂)

d(t)= s₁t + s₂t = (15 + s₂)t + s₂t = 15t + s₂t + s₂t = 15t + 2s₂t

We are told that at time t = 3 hours, d = 315 miles

d(3) = 15(3) + 2s₂(3) = 45 + 6s₂ = 315

Solve this for s₂

s₂ = 45 miles/hour

So now all we need is to find s₁

s₁ = 15 + s₂ = 15 + 45 = 60 miles/hour

Answer 2

Let r₁ and r₂ be the two bus rates in mph.

3r₁ + 3r₂ = 315

r₂ = r₁+15

3r₁ + 3(r₁+15)=315

3r₁ + 3r₁ + 45 = 315

6r₁ = 315-45=270

r₁ = 270/6 = 45 mph

r₂ = r₁+15 = 45 + 15 = 60 mph