Answer: the average speed of the boat relative to the water is 9 mph
Step-by-step explanation:
Let x represent the speed of the boat.
The rate of the current was 7 mph,
Assuming the boat travelled against the current while going upstream, the total speed would be x - 7
Assuming the boat in the direction of the current while going downstream, the total speed would be x + 7
Time = speed × time
Since she travelled 32 miles upstream and 32 miles downstream, then,
Time taken to go upstream = 32/(x -7)
Time taken to go downstream = 32/(x + 7)
Since the total time of the trip was 18 hours, then
32/(x -7) + 32/(x + 7) = 18
Cross multiplying by (x - 7)(x + 7), it becomes
32(x + 7) + 32(x - 7) = 18[(x - 7)(x + 7)
32x + 224 + 32x - 224 = 18(x² + 7x - 7x - 49
32x + 32x = 18x² - 882
18x² - 64x - 882 = 0
Dividing through by 2, it becomes
9x² - 32x - 441 = 0
9x² + 49x - 81x - 441 = 0
x(9x + 49) - 9(9x + 49) = 0
x - 9 = 0 or 9x + 49 = 0
x = 9 or x = - 49/9
Since the speed of the boat cannot be negative, then x = 9
