Rowing. Abby rows 10km. upstream and 10km. back in a total time of three hours. The speed of the river is 5km./h. Find Abby's speed in still water.
I just answered a very similar question here:
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Study the solution and see if you can apply it to your question.
Also I just noticed that the same question was answered here:
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All you have to do is solve the equation at the end.
can you please help find the solution to this equation. Your help would be greatly appreciated
To solve this problem, we can let "x" be Abby's speed in still water.
When Abby rows upstream against the current, her effective speed will be decreased by the speed of the river. So, her speed upstream will be (x - 5) km/h.
On the other hand, when Abby rows downstream with the current, her effective speed will be increased by the speed of the river. So, her speed downstream will be (x + 5) km/h.
Given that Abby rows 10 km upstream and 10 km back downstream in a total time of three hours, we can set up the following equation based on the formula time = distance / speed:
10 / (x - 5) + 10 / (x + 5) = 3
Now, let's solve this equation to find Abby's speed in still water (x):
First, we can simplify the equation by finding the common denominator:
(10(x + 5) + 10(x - 5)) / ((x - 5)(x + 5)) = 3
Expanding the numerator:
(10x + 50 + 10x - 50) / ((x - 5)(x + 5)) = 3
Combining like terms:
20x / ((x - 5)(x + 5)) = 3
Cross-multiplying:
20x = 3((x - 5)(x + 5))
Expanding the right side:
20x = 3(x^2 - 25)
Simplifying further:
20x = 3x^2 - 75
Rearranging the equation to quadratic form:
3x^2 - 20x - 75 = 0
Now, we can either factor this quadratic equation or use the quadratic formula to solve for x. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values from our quadratic equation:
x = (-(-20) ± √((-20)^2 - 4(3)(-75))) / (2(3))
Simplifying further:
x = (20 ± √(400 + 900)) / 6
x = (20 ± √1300) / 6
Now, let's calculate both possible values for x:
x₁ = (20 + √1300) / 6 ≈ 8.85 km/h
x₂ = (20 - √1300) / 6 ≈ -1.18 km/h
Since the speed of a boat cannot be negative, we can discard the negative value. Therefore, Abby's speed in still water is approximately 8.85 km/h.