Salmon hatch in freshwater streams and then migrate 70 miles downstream to the ocean in 8 hours. After reaching spawning age, they return upstream to fresh water to spawn and usually die soon after. If the salmon travel 58 miles in 8 hours during the upstream trip, find the rate of the salmons’ swimming in still water and the rate of the current?
if the salmon's speed is s, and the current's speed is c, then
since distance = speed * time,
(s+c)*8 = 70
(s-c)*8 = 58
now solve as usual
To find the rate of the salmon's swimming in still water and the rate of the current, we can use the concept of relative velocity.
Let's assume the rate of the salmon's swimming in still water is "x" miles per hour and the rate of the current is "c" miles per hour.
During the downstream journey, the salmon's effective velocity is the sum of its swimming speed and the speed of the current, i.e., (x + c) miles per hour.
Given that the salmon migrates 70 miles downstream in 8 hours, we can set up the equation as follows:
(distance) = (velocity) × (time)
70 = (x + c) × 8
Similarly, during the upstream journey, the salmon's effective velocity is the difference between its swimming speed and the speed of the current, i.e., (x - c) miles per hour.
Given that the salmon travels 58 miles upstream in 8 hours, we can set up the second equation as follows:
(distance) = (velocity) × (time)
58 = (x - c) × 8
Now we have a system of simultaneous equations:
1) 70 = (x + c) × 8
2) 58 = (x - c) × 8
To solve this system, we can rearrange the equations and use substitution or elimination method. Let's use substitution method:
From equation 1, we can rearrange it to x + c = 70/8 = 8.75. (dividing both sides by 8)
From equation 2, we can rearrange it to x - c = 58/8 = 7.25. (dividing both sides by 8)
Now we have two equations:
1) x + c = 8.75
2) x - c = 7.25
Adding the two equations:
(x + c) + (x - c) = 8.75 + 7.25
2x = 16
x = 16/2
x = 8
Therefore, the rate of the salmon's swimming in still water is 8 miles per hour.
Substituting the value of x into equation 1:
8 + c = 8.75
c = 8.75 - 8
c = 0.75
Therefore, the rate of the current is 0.75 miles per hour.
In conclusion, the rate of the salmon's swimming in still water is 8 miles per hour, and the rate of the current is 0.75 miles per hour.
To find the rate of the salmon's swimming in still water and the rate of the current, we can use the concept of relative speed.
Let's denote the rate of the salmon's swimming in still water as 's' and the rate of the current as 'c.'
During the downstream trip, the salmon are helped by the current, so their effective speed will be the sum of their swimming speed and the current speed. Therefore, we can write the equation:
Speed downstream = Swimming speed + Current speed
70 miles / 8 hours = s + c
Similarly, during the upstream trip, the salmon are swimming against the current, so their effective speed will be the difference between their swimming speed and the current speed. We can write the equation:
Speed upstream = Swimming speed - Current speed
58 miles / 8 hours = s - c
Now, we have two equations with two unknowns (s and c). We can solve these equations simultaneously to find the values.
Let's solve the first equation for s + c:
Speed downstream = s + c
70 miles / 8 hours = s + c
8s + 8c = 70
8s = 70 - 8c
s = (70 - 8c) / 8
Now, let's substitute this value of s in the second equation:
58 miles / 8 hours = s - c
58 / 8 = [(70 - 8c) / 8] - c
58 = 70 - 8c - 8c
16c = 70 - 58
16c = 12
c = 12 / 16
c = 0.75
Now, substitute the value of c in the equation for s:
s = (70 - 8c) / 8
s = (70 - 8 * 0.75) / 8
s = 69.5 / 8
s = 8.68
Therefore, the rate of the salmon's swimming in still water is approximately 8.68 miles per hour, and the rate of the current is approximately 0.75 miles per hour.