A motorboat travels the distance from one pier to another pier in 4 hours and the way back in 5 hours. What is the speed of the boat in still water if it travels 70 km with the current in 3.5 hours?
Let's call the distance between the piers "d". We can set up two equations based on the information given:
d = r1 * 4 (where r1 is the speed of the boat in still water on the way there)
d = r2 * 5 (where r2 is the speed of the boat in still water on the way back)
We can solve for "d" in both equations and set them equal to each other:
r1 * 4 = r2 * 5
r1 = (5/4) * r2
Now let's use the information about the boat traveling 70 km with the current in 3.5 hours. We know that the boat's speed relative to the water (which we'll call "c") is:
c = d / t
c = 70 / 3.5
c = 20
We can also set up an equation for the boat's speed with the current (which we'll call "b"), based on the speed in still water (r1):
b = r1 + c
Now we can substitute the expression for r1 from earlier into this equation:
b = (5/4) * r2 + 20
So now we have two expressions for the boat's speed: one in terms of r2, and one in terms of b. We can set them equal to each other and solve for r2:
(5/4) * r2 + 20 = b
(5/4) * r2 = b - 20
r2 = (4/5) * (b - 20)
Now we can substitute this expression for r2 back into the earlier equation:
r1 = (5/4) * r2
r1 = (5/4) * (4/5) * (b - 20)
r1 = b - 20
So we have an expression for r1 in terms of b. Finally, we can substitute this expression and the earlier expression for c into the first equation we set up:
d = r1 * 4
d = (b - 20) * 4
And now we can solve for b:
d = c * t
(b - 20) * 4 = 70
b - 20 = 17.5
b = 37.5
So the boat's speed in still water is 37.5 km/h.
speed in still water = s
speed with current = s+c
speed against current = s-c
==========================
(s+c) (3.5) = 70
(s+c) * 4 = d the distance between piers
(s-c) * 5 = d
=======================
4 s + 4 c = d
5 s - 5 c = d
so
-s + 9 c = 0
c = s/9
but we know
s + c = 70/3.5
s + s/9 = 20
10 s / 9 = 20
s = 180/10 = 18
Therefore, the speed of the boat in still water is 18 km/h.
That is what I said. However it is not what you said.
I apologize for the confusion and any mistake I might have made in my explanation. Thank you for providing the correct solution.
To find the speed of the boat in still water, we need to break down the given information and use it to solve for the boat's speed.
Let's first assume that the distance between the piers is denoted by "d" kilometers.
From the information provided, we know that:
- The boat travels from one pier to another pier in 4 hours.
- The boat travels back from the second pier to the first pier in 5 hours.
- The boat travels 70 km with the current in 3.5 hours.
Now, let's consider the speed of the boat in still water, denoted by "b" km/h, and the speed of the current, denoted by "c" km/h.
When the boat is traveling from one pier to another:
- The effective speed of the boat when moving with the current is (b + c) km/h.
- The effective speed of the boat when moving against the current is (b - c) km/h.
Using the formula: speed = distance/time, we can create two equations:
Equation 1: (b + c) = d/4 (since distance/speed = time)
Equation 2: (b - c) = d/5
Now, let's solve these equations to find the value of "b" (the speed of the boat in still water).
To eliminate "c", we can add Equation 1 and Equation 2:
(b + c) + (b - c) = d/4 + d/5
2b = (5d + 4d)/(4*5)
2b = (9d/20)
b = (9d/20) * (1/2)
b = 9d/40
Now, we have an expression for "b" in terms of "d".
To find the value of "d", we can use the third piece of given information: the boat travels 70 km with the current in 3.5 hours.
Using the equation: speed = distance/time, we can get another equation:
70 = (b + c) * 3.5
70 = ((9d/40) + c) * 3.5
70 = (9d/40 + c) * 7/2
70 = (9d + 20c)/40 * 7/2
70 * 80/7 = 9d + 20c
To solve for "d", we need to find the values of "b" and "c". We can substitute the expression for "b" we derived earlier into this equation, but we don't have enough information to solve for "c" directly.
Therefore, we cannot determine the exact speed of the boat in still water with the given information.