Quadratic Application

Question

A car travels 780 mi at a certain speed. If the speed had been 5 mph faster, the trip would have been made in 1 hr less time. Find the speed.

Answer 1

We'll let v be the speed of the car in mph, and t be the time it takes in hours to travel 780 miles. 

 

The distance traveled is always v*t, so we have

 

780=v*t

 

If the speed had been 5 miles per hour faster, the time to travel this same 780 miles would be one hour less:

 

780=(v+5)(t-1)

 

These are the two equations we need to solve. We'll eliminate t since v is what we ultimately need to find. Start by solving both equations for t:

 

t=780/v

 

and

 

t=780/(v+5)+1

 

We can set these terms equal to each other since they are both equal to t:

 

780/v=780/(v+5)+1

 

Now, we can multiply both sides of the equation by v(v+5). This gets rid of the fractions:

 

780(v+5)=780v+v(v+5)

 

Distributing through the parentheses on both sides:

 

780v+3900=780v+v²+5v

 

We can cancel 780v from both sides:

 

3900=v²+5v

 

Now, bring everything to the same side by subtracting 3900 from both sides:

 

0=v²+5v-3900

 

Now we need to factor this quadratic. We need to numbers that multiply to -3900 and add to 5. It is not easy to see the factorization, but the correct one is

 

0=(v-60)(v+65)

 

If you didn't see that immediately, you can also use the quadratic formula:

 

v=(-5±√(25+4*(-3900)))/2=(-5±√(25+15600))/2=(-5±√15625)/2=(-5±125)/2={-130/2,120/2}={-65,60}

 

Since the speed is a positive number, only v=60 makes sense. That is the answer.