A motorboat can maintain a constant speed of 16 miles per hour relative to the water. The boat makes a trip upstream to a certain point in 20 minutes and the return trip takes 15 minutes. What is the speed of the current?
To solve this problem, we can use the concept of relative velocity. Let's assume the speed of the current is "S" miles per hour.
When the motorboat is moving upstream, it is moving against the current. In this case, the effective speed of the boat will be reduced by the speed of the current. So the relative speed of the boat with respect to the ground will be 16 - S miles per hour.
Similarly, when the boat is moving downstream, it is moving with the current. In this case, the effective speed of the boat will be increased by the speed of the current. So the relative speed of the boat with respect to the ground will be 16 + S miles per hour.
Now let's use the formula: distance = speed × time.
On the upstream trip, the boat takes 20 minutes (or 20/60 = 1/3 hours) to cover the distance. Let's denote the distance as "D".
So, we have the equation: D = (16 - S) × (1/3).
On the downstream trip, the boat takes 15 minutes (or 15/60 = 1/4 hours) to cover the same distance. Again, using the formula, we have: D = (16 + S) × (1/4).
Since both equations refer to the same distance, we can equate them: (16 - S) × (1/3) = (16 + S) × (1/4).
Simplifying this equation, we have: (16 - S) × 4 = (16 + S) × 3.
Expanding both sides, we get: 64 - 4S = 48 + 3S.
Moving all the "S" terms to one side, we have: 7S = 64 - 48.
Simplifying further, we get: 7S = 16.
Dividing both sides by 7, we find: S = 16/7.
Therefore, the speed of the current is approximately 2.29 miles per hour.