A person rowed her boat upstream a distance of 45
mi and then rowed back to the starting point. The total time of the trip was 14
hours. If the rate of the current was 2
mph, find the average speed of the boat in still water.
since time = distance/speed,
45/(x-2) + 45/(x+2) = 14
To find the average speed of the boat in still water, we need to consider the speed of the boat relative to the water and the speed of the current.
Let's assume the speed of the boat in still water is B mph. Since the boat is rowing upstream, its effective speed is reduced by the speed of the current.
The speed of the boat relative to the water when rowing upstream is (B - 2) mph.
The time taken to row upstream a distance of 45 miles at a speed of (B - 2) mph is given by the formula:
Time = Distance / Speed
Time taken upstream = 45 / (B - 2)
When rowing downstream, the speed of the boat relative to the water is increased by the speed of the current, so it becomes (B + 2) mph.
The time taken to row back downstream a distance of 45 miles at a speed of (B + 2) mph is given by the formula:
Time = Distance / Speed
Time taken downstream = 45 / (B + 2)
Since the total time for the trip was 14 hours, we can set up the equation:
Time taken upstream + Time taken downstream = 14
45 / (B - 2) + 45 / (B + 2) = 14
To solve this equation, we can multiply both sides by (B - 2)(B + 2) to eliminate the denominators:
45(B + 2) + 45(B - 2) = 14(B - 2)(B + 2)
Simplifying the equation:
45B + 90 + 45B - 90 = 14(B^2 - 4)
Combine like terms:
90B = 14B^2 - 56
Rearrange the equation into a quadratic form:
14B^2 - 90B - 56 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. Once we find the value(s) of B, we can determine the average speed of the boat in still water.