A motorboat traveling with the current can go 160 km in 4 h. Against the current it takes 5 h to go the same distance. Find the rate of the motorboat in still water and the rate of the current.
Please help!
To find the rate of the motorboat in still water and the rate of the current, we need to set up two equations based on the given information.
Let's assume the rate of the motorboat in still water is represented by "b" (in km/h), and the rate of the current is represented by "c" (in km/h).
When the motorboat is traveling with the current, the effective speed of the boat is increased by the speed of the current. Therefore, the equation for this scenario can be written as:
Distance = Speed * Time
160 km = (b + c) km/h * 4 h
Similarly, when the motorboat is traveling against the current, the effective speed is decreased by the speed of the current. Therefore, the equation for this scenario can be written as:
Distance = Speed * Time
160 km = (b - c) km/h * 5 h
Now we have a system of two equations:
1) 160 = (b + c) * 4
2) 160 = (b - c) * 5
To solve this system, we can use the method of substitution or elimination. Let's use the elimination method:
Multiplying equation 1 by 5 and equation 2 by 4, we get:
1) 5 * 160 = 4 * (b + c)
2) 4 * 160 = 5 * (b - c)
Simplifying, we have:
1) 800 = 4b + 4c
2) 640 = 5b - 5c
Now, let's add these two equations together:
800 + 640 = 4b + 4c + 5b - 5c
Simplifying, we have:
1440 = 9b - c
Since we want to find both b and c, we need another equation. Fortunately, we have the equation for the speed of the boat in still water (b):
Distance = Speed * Time
160 km = b km/h * 4 h
Simplifying, we have:
4b = 160
Dividing both sides by 4, we find:
b = 40
Now we can substitute the value of b in equation 1440 = 9b - c:
1440 = 9(40) - c
Simplifying further:
1440 = 360 - c
Adding c to both sides:
c + 1440 = 360
Subtracting 1440 from both sides:
c = 1440 - 360
Simplifying:
c = 1080
Therefore, the rate of the motorboat in still water is 40 km/h, and the rate of the current is 1080 km/h.
b + c = 160 / 4
b - c = 160 / 5
add the two equations (cancelling c)
substitute back to find c