Quadratic - speed, distance, time

Question

On a journey of 100km, Jim travels at a certain speed for the first 60km, and then increases his speed by 15km/h for the remainder. If the whole journey takes 2 hours, find his two speeds.

Answer 1

Distance = (Rate)(Time), so Time = Distance/Rate

 

Total time = 2 hours

 

Let x = speed for the first 60 km

 

Then, x + 15 = speed for the last 40 km

 

Time at slower speed + Time at faster speed = Total time

 

60/x + 40/(x+15) = 2

 

Divide both sides by 2:  30/x + 20/(x+15) = 1

 

Multiply by x(x+15):  30(x+15) + 20x = x(x+15)

 

Simplify to get 50x + 450 = x² + 15x

 

So, x² - 35x - 450 = 0

 

      (x - 45)(x + 10) = 0

 

Therefore, x = 45  or x = -10

 

Since x = speed, x can't be negative.  

 

So, x = 45 kph and x + 15 = 60 kph

Answer 2

Speed, s, is the quantity distance, d, over time, t.

So, set up your equation using the information given:

 

let, (2-t), represent the time it took Jim to travel in the increased speed

and (s+15) is the increased speed Jim traveled by for the remainder

1. 100= 60 + 40
2. s = 60/t
3. (s+15)=40/(2-t)
4. Plug in  #2 in #3,     (60/t) +15 = 40/(2-t)
5. Simplify*, 3t² + 14t - 24= 0
6. Solve, t= 4/3 or 1 hour and 20 minutes

So, plug in #2, 

s = 60/(4/3)

s = 45 km/hr, the speed with which Jim traveled the first 60km

s+15= 60 km/hr, the speed for the remainder 40 km.

 

Note: * To simplify, multiply both sides of the equation by t(2-t), the least common multiple of the denominators also known as greatest common denominator. Then combine like terms and collect all the terms to one side of the equation.