Simplify the sum. state any restrictions on the variables.
(x + 6)/x^2 - 36 + 1/x - 6
2. The speed of the current in a river is 4 mph. A ferry operator who works that
part of the river is looking to buy a new boat for his business. Every day, his
route takes him 24 miles each way against the current and back to his dock,
and he needs to make this trip in a total of 8 hours. He has a boat in mind, but
he can only test it on a lake where there is no current. How fast must the boat
go on the lake in order for it to serve the ferry operator's needs? (Hint: Use time
= distance/speed to start)
1. Simplified sum:
(x + 6)/x^2 - 36 + 1/x - 6
= (x + 6)/(x - 6)(x + 6) + 1/x - 6
= 1/(x - 6) + 1/x - 6
= (x + 1 - 6x(x - 6))/(x(x - 6))
= (-6x^2 + 35x - 6)/(x(x - 6))
Restriction on variables: x cannot be equal to 0 or 6.
2. Let the speed of the boat in still water be b mph.
Against the current: time = distance/speed
Time taken = 24/(b - 4)
With the current: time = distance/speed
Time taken = 24/(b + 4)
Total time taken = 8 hours
24/(b - 4) + 24/(b + 4) = 8
24(b + 4) + 24(b - 4) = 8(b^2 - 16)
48b = 8b^2 - 128
8b^2 - 48b - 128 = 0
b^2 - 6b - 16 = 0
(b - 8)(b + 2) = 0
b = 8 or b = -2
Since speed cannot be negative, the boat must go at a speed of 8 mph in still water.