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 7.0 mi/h, the time to go one mile decreases by 14 s. What was your original speed?
Please explain me step by step
Let's denote the original speed as x in mi/h.
We are given that it takes a time t to go from one mile marker to the next at this speed.
When the speed is increased by 7.0 mi/h, the new speed becomes (x + 7.0) mi/h.
We are also given that the time to go one mile decreases by 14 seconds at this new speed.
Let's consider the time it took to travel one mile initially at the original speed. This can be represented as:
Distance = Speed * Time
1 mile = x * t
Now, let's consider the time it takes to travel one mile at the increased speed. This can be represented as:
1 mile = (x + 7.0) * (t - 14)
We can rewrite this equation as:
x * t - 14x + 7.0t - 98 = x * t
Simplifying the equation, we get:
-14x + 7.0t - 98 = 0
Now, let's isolate x to solve for it:
-14x = -7.0t + 98
Dividing both sides by -14, we have:
x = (7.0t - 98) / 14
Simplifying further, we get:
x = 0.5t - 7
Therefore, the original speed was 0.5t - 7 mi/h.
To find the original speed, we can set up a system of equations using the given information. Let's denote the original speed as v mi/h.
From the problem, we know that the time it takes to travel one mile at the original speed is t seconds. Therefore, the original speed can be defined as:
v = 1 mile / t (equation 1)
When the speed is increased by 7.0 mi/h, the new speed becomes (v + 7.0) mi/h. The time it takes to travel one mile at this new speed is (t - 14) seconds. So, the new speed can be defined as:
v + 7 = 1 mile / (t - 14) (equation 2)
Now, we have a system of two equations with two variables (v and t). We can solve this system to find the values of v and t.
Let's rearrange equation 1 to solve for t:
t = 1 mile / v
Substituting this value of t in equation 2, we get:
v + 7 = 1 mile / (1 mile / v - 14)
Now, we can simplify this equation by multiplying both sides by (1 mile / v - 14):
(v + 7)(1 mile / v - 14) = 1 mile
Expanding the left side of the equation:
(v + 7) / v - 14(v + 7) = 1
Now, we can distribute -14 to the terms inside the parenthesis:
(v + 7) / v - 14v - 98 = 1
Next, we combine the terms on the left side of the equation:
(v + 7) - 14v - 98v = v
Simplify:
v + 7 - 14v - 98v = v
-112v - 7 = 0
Rearranging this equation:
-112v = 7
Dividing both sides by -112, we get:
v = 7 / -112
Calculating this value, we get:
v ≈ -0.0625
However, it does not make physical sense to have a negative speed. Therefore, we discard this negative value and conclude that the original speed is v ≈ 0.0625 mi/h.
So, your original speed was approximately 0.0625 mi/h.