The current in a stream moves at a speed of 9 mph. A boat travels 17 mi upstream and 17 mi downstream in a total time of 13 hrs. What is the speed of the boat in still water?
speed in still water ---- x
rate upstream = x-9 , where x is clearly > 9
rate downstream = x+9
17/(x+9) + 17/(x-9) = 13
times (x+)(x=9) or x^2 - 81
17(x-9) + 17(x+9) = 13(x^2 - 81)
17x - 153 + 17x + 153 = 13x^2 - 1053
13x^2 - 34x - 1053 = 0
x = (34 ± √55912)/26
= appr 10.4 or a negative which would make no sense.
the boat can go 10.4 mph in still water
To solve this problem, we can use the concept of relative velocity.
Let's assume the speed of the boat in still water is 'b' mph, and the speed of the current is 'c' mph.
When the boat is moving upstream, its effective speed is the difference between the boat's speed in still water and the speed of the current: (b - c) mph.
Similarly, when the boat is moving downstream, its effective speed is the sum of the boat's speed in still water and the speed of the current: (b + c) mph.
Now, let's calculate the times taken for upstream and downstream journeys:
Time taken for upstream journey = Distance / Speed = 17 miles / (b - c) mph
Time taken for downstream journey = Distance / Speed = 17 miles / (b + c) mph
According to the problem, the total time taken for both journeys is 13 hours:
Time taken for upstream journey + Time taken for downstream journey = 13 hours
Substituting the calculated times, we have:
17 / (b - c) + 17 / (b + c) = 13
To solve this equation, we can multiply through by (b - c)(b + c) to eliminate the denominators:
17(b + c) + 17(b - c) = 13(b - c)(b + c)
Expanding and simplifying, we get:
17b + 17c + 17b - 17c = 13(b^2 - c^2)
Combining like terms, we have:
34b = 13b^2 - 13c^2
Rearranging the equation, we get:
13b^2 - 34b - 13c^2 = 0
This is a quadratic equation in terms of 'b'. Solving this equation will give us the speed of the boat in still water.