the barge travels upstream 755 miles to the factory and then

the barge travels upstream 75.5 miles to the factory and then back to the warehouse.the barge travels upstream at a rate of 5 miles per hour slower than it travels downstream.the barge total time is 15 hours.what is the rate the barge travels downstream

Solution

distance for travelling one way (d) = 75.5 miles

let downstream speed is x miles/hr

then upstream speed will be (x-5) miles/hr

we know time = distance/time

given total time taken for journey = 15 hrs

                                                 = upstream time + downstream time

                                                  = (75.5/x) + (75.5/(x-5))

=> 15/75.5 = 1/x + 1/(x-5)

=>30/151 = 1/x   + 1(x-5)

=>30/151 = (x-5 +x)/x(x-5)

=>30* (x² -5x) = 151*(2x-5)

=> 30x² -150x =     302x-755

=>30x² -452x +755 = 0

solving we get x = {452 + - sqrt(452² - 4*30 *755)}/4*30

                       ={452 + - 337.200237248}/120

=>x= 0.9566646896 miles/hr or x = 6.57666864373 miles/hr

but for x = 0.9566646896 , (x-5) = -4.0433353104, negative upstream speed can\'t be possible

so only feasible solution is x = 6.57666864373 miles/hr, it is the rate at which the barge travels downstream