a boat moves through still wather at x kilometers(km) per hour. it travels 24km up stream against a current of 2km per hour and then returns with the current. The rate upstream is x-2 because the boat moves against the current and the rate downstream is x+2 because the boat moves with the current.
36.write an expression for the total time for the round trip.
37.write your answer to exercise 36 as a single rotional expression.
28.use your answer to exercise 37 to find how long the round trip will take if the boat travels 10kilometers per hour through still water
time upstream = 24/(x-2)
time downstream = 24/(x+2)
total time = 24/(x-2) + 24/(x+2)
common denominator is (x+2)(x-2)
so total = (24(x+2) + 24(x-2))/((x+2)(x-2))
= 48x/(x^2 - 4)
Plug in x=10
36. To calculate the total time for the round trip, we need to consider the time it takes to travel upstream and the time it takes to travel downstream.
The time it takes to travel upstream can be found by dividing the distance by the rate upstream, which is 24 km / (x - 2) km/h.
The time it takes to travel downstream can be found by dividing the distance by the rate downstream, which is 24 km / (x + 2) km/h.
To calculate the total time for the round trip, we add these two times together:
Total time = Time upstream + Time downstream
Total time = 24 / (x - 2) + 24 / (x + 2)
37. Expressing the answer to exercise 36 as a single rational expression involves finding a common denominator and combining the fractions:
Total time = (24(x + 2) + 24(x - 2)) / (x - 2)(x + 2)
Total time = (24x + 48 + 24x - 48) / (x^2 - 4)
Total time = 48x / (x^2 - 4)
28. To find how long the round trip will take if the boat travels at 10 km/h through still water, we substitute x = 10 into the expression from exercise 37:
Total time = 48(10) / (10^2 - 4)
Total time = 480 / (100 - 4)
Total time = 480 / 96
Total time = 5 hours