Cities A, B, C and D are equidistant. That is, B is d miles from A, C is d miles from B, and D is d miles from C. Suppose you travel 55 mph from A to B, 40 mph from B to C, and 50 Mph from C to D. What single, constant speed could you have traveled from A to B to C to D that would have taken the same amount of time?
I know that I am supposed to use harmonic mean. Just not sure how to set up formula.
To solve this problem using the harmonic mean, let's first determine the time it takes to travel from A to B, from B to C, and from C to D separately.
Given:
Speed from A to B = 55 mph
Speed from B to C = 40 mph
Speed from C to D = 50 mph
Let's assume the distance between consecutive cities is d miles.
Time to travel from A to B = Distance / Speed = d / 55 hours
Time to travel from B to C = Distance / Speed = d / 40 hours
Time to travel from C to D = Distance / Speed = d / 50 hours
Now, we need to find a single, constant speed that would have taken the same amount of time for the entire journey A to B to C to D. Let's denote this speed as S mph.
We can use the harmonic mean to find this speed. The harmonic mean of three values is defined as the reciprocal of the arithmetic mean of their reciprocals.
Reciprocal of the time to travel from A to B: 1 / (d / 55) = 55 / d
Reciprocal of the time to travel from B to C: 1 / (d / 40) = 40 / d
Reciprocal of the time to travel from C to D: 1 / (d / 50) = 50 / d
Now, we can find the harmonic mean of these reciprocals:
Harmonic Mean = 3 / [(1 / (d / 55)) + (1 / (d / 40)) + (1 / (d / 50))]
= 3 / [(55 / d) + (40 / d) + (50 / d)]
= 3 / [(145 / d)]
= 3d / 145
Therefore, the single, constant speed that would have taken the same amount of time for the entire journey A to B to C to D is given by (3d / 145) mph.