A river flows at the rate of 2 mph. If a motorboat can travel 12 miles upstream and return in a total of 2.5 hours, what is the speed of the motorboat in still water?
d / r = t
[12 / (m - 2)] + [12 / (m + 2)] = 2.5
12(m + 2) + 12(m - 2) = 2.5(m^2 - 4)
0 = 2.5 m^2 - 24 m - 10
use quadratic formula to find m
To find the speed of the motorboat in still water, we can use the concept of relative speed.
Let's assume the speed of the motorboat in still water is x mph.
When the motorboat is traveling upstream, against the current, its effective speed will be (x - 2) mph (as the river flows at a rate of 2 mph in the opposite direction).
When the motorboat is traveling downstream, with the current, its effective speed will be (x + 2) mph.
Given that the motorboat can travel 12 miles upstream and return in a total of 2.5 hours, we can set up the following equation:
12/(x - 2) + 12/(x + 2) = 2.5
To solve this equation, we can multiply both sides by (x - 2)(x + 2) to eliminate the denominators:
12(x + 2) + 12(x - 2) = 2.5(x - 2)(x + 2)
Simplifying this equation gives us:
12x + 24 + 12x - 24 = 2.5(x^2 - 4)
Combine like terms:
24x = 2.5x^2 - 10
Rearrange the equation to put it in quadratic form:
2.5x^2 - 24x - 10 = 0
Now, we can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values, a = 2.5, b = -24, and c = -10, we get:
x = (-(-24) ± √((-24)^2 - 4*(2.5)*(-10))) / (2*(2.5))
Simplifying further:
x = (24 ± √(576 - (-100))) / 5
x = (24 ± √(576 + 100)) / 5
x = (24 ± √676) / 5
x = (24 ± 26) / 5
Now we have two possible values for x:
x1 = (24 + 26) / 5 = 50 / 5 = 10 mph
x2 = (24 - 26) / 5 = -2 / 5 = -0.4 mph
Since the speed of the motorboat cannot be negative, we can conclude that the speed of the motorboat in still water is 10 mph.
To find the speed of the motorboat in still water, we first need to find the speed of the current.
Let's denote the speed of the motorboat in still water as "x" mph.
When the motorboat travels upstream (against the current), its effective speed is reduced by the speed of the current. Thus, the effective speed is (x - 2) mph.
Similarly, when the motorboat travels downstream (with the current), its effective speed is increased by the speed of the current. The effective speed is (x + 2) mph.
Now we can use the formula time = distance / speed to set up the equations for the upstream and downstream trips.
For the upstream trip:
time = distance / speed
2.5 - (12 / (x - 2)) = distance / (x - 2)
For the downstream trip:
time = distance / speed
2.5 - (12 / (x + 2)) = distance / (x + 2)
By solving these two equations simultaneously, we can find the value of x, which represents the speed of the motorboat in still water.
To solve these equations, you can use substitution or elimination methods.
Well, since the river is flowing at 2 mph and the boat can travel 12 miles upstream, that means it takes the boat 6 hours to go 12 miles against the current.
Now, if we add the time it takes to return downstream, we have a total of 6 hours + 6 hours = 12 hours.
But wait, we were told that the total time is actually 2.5 hours. That doesn't seem right! It looks like the boat has found a way to bend the space-time continuum!
In all seriousness though, something doesn't add up here. It seems like we have some incorrect information or some mathematical trickery going on. So, unfortunately, I can't give you an answer to this riddle.