Driving along a crowded freeway, you notice that it takes a time t to go from one mile marker to the next. When you increase your speed by 4.5 mi/h, the time to go one mile decreases by 12 s. What was your original speed?
To solve this problem, we can set up two equations based on the information given.
Let's start by defining the original speed as "s" mi/h.
According to the problem, it takes time "t" to travel one mile at the original speed.
So, the first equation we can write is:
t = 1 mile / s mi/h
Next, we are given that when the speed is increased by 4.5 mi/h, the time to travel one mile decreases by 12 seconds. This means the new time to travel one mile is (t - 12) seconds.
So, the second equation we can write is:
(t - 12) seconds = 1 mile / (s + 4.5) mi/h
Now, we have a system of two equations:
t = 1 / s (equation 1)
t - 12 = 1 / (s + 4.5) (equation 2)
To solve this system of equations, we can use substitution. Let's solve equation 1 for "t" and substitute it into equation 2:
t = 1 / s
(t - 12) = 1 / (s + 4.5)
Substituting the value of t from equation 1 into equation 2, we get:
(1 / s) - 12 = 1 / (s + 4.5)
To simplify this equation further, we can multiply both sides by s(s + 4.5) to get rid of the denominators:
s(s + 4.5)((1 / s) - 12) = s(s + 4.5)(1 / (s + 4.5))
After simplifying, we get:
s + 4.5 - 12s = s
Simplifying further, we have:
-11s + 4.5 = 1
Moving the terms around, we get:
-11s = 1 - 4.5
-11s = -3.5
Finally, we can solve for s by dividing both sides of the equation by -11:
s = (-3.5) / (-11)
s = 3.5 / 11
s ≈ 0.318 (rounded to three decimal places)
Therefore, your original speed was approximately 0.318 mi/h.