How do I set up an equation to solve this? Thanks for any help you can give me!

Question

Driving along a crowded freeway, you notice that it takes a time "t" to go from on mile marker to the next. When you increse your speed by 7.8 mph, the time to go one mile decreases by 14 seconds. What was your original speed?

Answer 1

Let's start with our basic equation relating time, speed, and distance:  d = st.

 

Going from one mile marker to the next is a distance of one mile, so d = 1.  Our equation for the initial situation is then just

 

1 = st, or t = 1/s

We don't know the speed or the time yet - we just know the relationship between them.

We now modify the speed, and observe that the time changes.  One important caveat, though!  Since we're measuring speed in miles per HOUR, we should turn the 14 second time decrease into hours as well, to make sure that our units agree with each other.  To convert from seconds to hours, we need to multiply by (1 hour/3600 seconds) - this gives us .00389 hours.

 

Our new equation is then

1 = (s+7.8)(t - .00389)

1 = st + 7.8t - .00389s - .0303

 

Substitute our original t, above:

 

1 = s(1/s) + 7.8/s - .00389s - .0303

1 = 1 + 7.8/s - .00389s - .0303

0 = 7.8/s - .00389s - .0303

 

Having terms over s isn't great - let's multiply everything by one power of s.

 

0 = 7.8 - .00389s² - .0303s

This is becoming recognizable as a quadratic function!

 -.00389s² - .0303s + 7.8 = 0

 

We can simply use the quadratic equation, now, to solve for s:

s = [.0303 ± √(.0303² + 4*.00389*7.8)]/(-.00788)

= [.0303 ± √0.122]/(-.00788)

~= [.0303 ± .350]/(-.00788)

Based on the logic of the situation, we want the positive answer, which will be the MINUS .350, since we're dividing by a negative number.

s = -0.319/(-.00788) ~= 40.5 mph.

 

Check to see that it works:  At 40.5 mph, the time between mile markers would be .0247 hours, or 88.9 seconds.  Accelerating by 7.8 mph would take you to 48.3 mph, at which point the time between mile markers would be .0207 hours, or 74.5 seconds, a difference of 14.4 - maybe I should have kept another digit in those decimals.  But it's certainly the right vicinity.