need help.
Young salmon hatch in freshwater steams and then migrate downstream to the ocean at the rate of 70 miles in 8 hours. there they live until they reach spawning age. then they return upstream to fresh water to spawn and usually die soon after. on the upstream trip, the salmon travel about 58 miles in 8 hours. find the rate of the salmon swimming in still water and the rate of the current.
this is what I have so far, but it doesnt add up
r*t=d
downstream= 8(x+y)=70
upstream=8(x-y)=58
OK, it looks like you have chosen x for the rate in still water and y for the speed of the current. The equations you should have written are:
x + y = 70/8
x - y = 58/8
which is the same as you have written.
Adding them gives
2x = 128/8 = 16
x = 6 mph
y = 70/8 - 6 = 2.75 mph
You started the problem correctly and only needed to finish.
To find the rate of the salmon swimming in still water (which we'll call "r") and the rate of the current (which we'll call "c"), we can set up a system of equations based on the information given.
Let's use the formula: distance = rate × time.
For the downstream trip:
Distance = 70 miles,
Time = 8 hours,
Rate = r + c (salmon swimming rate plus current speed).
So, the equation for the downstream trip is:
70 = 8(r + c).
For the upstream trip:
Distance = 58 miles,
Time = 8 hours,
Rate = r - c (salmon swimming rate minus current speed).
Therefore, the equation for the upstream trip is:
58 = 8(r - c).
Now, we have a system of two equations:
1) 70 = 8(r + c),
2) 58 = 8(r - c).
To solve this system of equations, we can use the method of substitution or elimination.
Let's use the elimination method to solve this system:
Multiply the first equation by 1/8 to simplify it:
(1/8) * 70 = (1/8) * 8(r + c),
8.75 = r + c.
Now, we can subtract the second equation from this simplified version of the first equation:
(8.75) - 58 = (r + c) - (r - c),
-49.25 = 2c.
Divide both sides of the equation by 2 to solve for c:
c = -49.25 / 2 = -24.625.
So, the rate of the current is -24.625 mph. The negative sign indicates that the current is flowing in the opposite direction of the salmon's swimming.
To find the rate of the salmon swimming in still water (r), we can substitute the value of c (-24.625) into the simplified version of the first equation:
8.75 = r + (-24.625).
8.75 = r - 24.625.
Add 24.625 to both sides:
8.75 + 24.625 = r - 24.625 + 24.625,
33.375 = r.
Therefore, the rate of the salmon swimming in still water is 33.375 mph.
To summarize:
- The rate of the salmon swimming in still water is 33.375 mph.
- The rate of the current is -24.625 mph.