If the river current flows at an average 3 miles per hour, then a tour boat makes the 9-mile tour downstream with the current and back the 9 miles against the current in 4 hours. What is the average speed of the boat in still water?
since time = distance/speed,
9/(s-3) + 9/(s+3) = 4
Let's denote the average speed of the boat in still water as "x" miles per hour.
When the boat is going downstream with the current, its effective speed is the sum of the boat's speed in still water and the speed of the current. So, the effective speed is x + 3 miles per hour.
Similarly, when the boat is going against the current upstream, its effective speed is the difference between the boat's speed in still water and the speed of the current. So, the effective speed is x - 3 miles per hour.
We know that the tour boat makes the 9-mile trip downstream and back upstream in a total of 4 hours. To solve for the average speed of the boat in still water, we can set up the following equation:
9/(x + 3) + 9/(x - 3) = 4
Now we can solve this equation step by step:
1. Multiply both sides of the equation by (x + 3)(x - 3) to eliminate the denominators:
9(x - 3) + 9(x + 3) = 4(x + 3)(x - 3)
2. Simplify and distribute:
9x - 27 + 9x + 27 = 4(x^2 - 9)
3. Combine like terms:
18x = 4x^2 - 36
4. Move all terms to one side to set the equation to zero:
4x^2 - 18x - 36 = 0
5. Divide the entire equation by 2 to simplify:
2x^2 - 9x - 18 = 0
Now we can solve this quadratic equation. We can either factor it or use the quadratic formula. Let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac))/(2a)
Using this formula with a = 2, b = -9, and c = -18, we have:
x = (-(-9) ± sqrt((-9)^2 - 4(2)(-18)))/(2(2))
x = (9 ± sqrt(81 + 144))/(4)
x = (9 ± sqrt(225))/(4)
x = (9 ± 15)/(4)
So we have two possible solutions for x:
x1 = (9 + 15)/4 = 6
x2 = (9 - 15)/4 = -1.5
Since the average speed of the boat cannot be negative, the answer is x = 6 miles per hour. Therefore, the average speed of the boat in still water is 6 miles per hour.
To find the average speed of the boat in still water, we can assign variables to the unknowns involved and apply the formula:
Let's assume the speed of the boat in still water is b miles per hour.
Given that the current flows at an average speed of 3 miles per hour, we can determine the effective speed of the boat when it is moving downstream (with the current) and upstream (against the current).
When the boat is moving downstream, the speed of the boat relative to the river current is:
b + 3 miles per hour
When the boat is moving upstream, the speed of the boat relative to the river current is:
b - 3 miles per hour
Now, let's calculate the time it takes for the boat to complete the tour.
Distance = Rate × Time
Downstream trip:
9 = (b + 3) × time_downstream
Upstream trip:
9 = (b - 3) × time_upstream
We know that the total time for the round trip is 4 hours.
So, the total time for the downstream and upstream trips combined is 4 hours.
time_downstream + time_upstream = 4
Now, we have a system of two equations:
1) 9 = (b + 3) × time_downstream
2) 9 = (b - 3) × time_upstream
3) time_downstream + time_upstream = 4
From equation 3, we can express time_downstream in terms of time_upstream:
time_downstream = 4 - time_upstream
Substitute this value into equation 1:
9 = (b + 3) × (4 - time_upstream)
Now, we can rewrite equation 2 in terms of time_upstream:
9 = (b - 3) × time_upstream
Simplify the equations:
4b - (b + 3) × time_upstream = 3 × time_upstream
9 = (b - 3) × time_upstream
Expand and rearrange the equations:
4b - 3 × time_upstream = 3 × time_upstream + 9
4b - 3 × time_upstream = 3 × time_upstream + 9
Combine like terms:
4b = 6 × time_upstream + 9
Now, substitute the value of time_upstream in terms of time_downstream from equation 3:
4b = 6 × (4 - time_downstream) + 9
Simplify the equation:
4b = 24 - 6 × time_downstream + 9
Combine like terms:
4b = 33 - 6 × time_downstream
Rearrange the equation for time_downstream:
6 × time_downstream = 33 - 4b
Divide both sides by 6:
time_downstream = (33 - 4b) / 6
Substitute this value of time_downstream into equation 1:
9 = (b + 3) × ((33 - 4b) / 6)
Now, we can solve this equation to find the value of b, the speed of the boat in still water.