A cyclist travels 70 miles in 7 hours going aganist the wind and 56 miles in 4 hours with the wind. what is the rate of the cyclist in still air and and the rate of the wind.
rate in still water=
rate of the wind=
To find the rate of the cyclist in still air and the rate of the wind, we can use a system of equations. Let's assume that the rate of the cyclist in still water is represented by "r" and the rate of the wind is represented by "w."
When the cyclist is traveling against the wind, the effective rate of the cyclist is reduced by the rate of the wind. So, the equation for this scenario is:
(r - w) * 7 = 70
When the cyclist is traveling with the wind, the effective rate of the cyclist is increased by the rate of the wind. So, the equation for this scenario is:
(r + w) * 4 = 56
Now, we can solve this system of equations to find the values of "r" and "w."
First, let's simplify the equations:
1. (r - w) * 7 = 70
7r - 7w = 70 (equation 1)
2. (r + w) * 4 = 56
4r + 4w = 56 (equation 2)
To eliminate the variable "w," let's multiply equation 2 by 7:
28r + 28w = 392 (equation 3)
Now, let's add equation 1 and equation 3:
7r - 7w + 28r + 28w = 70 + 392
35r = 462
Divide both sides by 35:
r = 462 / 35
r = 13.2
So, the rate of the cyclist in still air is 13.2 mph.
To find the rate of the wind, substitute the value of "r" into equation 1:
7(13.2) - 7w = 70
92.4 - 7w = 70
Now, solve for "w":
-7w = 70 - 92.4
-7w = -22.4
Divide both sides by -7:
w = -22.4 / -7
w = 3.2
So, the rate of the wind is 3.2 mph.
Therefore, the rate of the cyclist in still air is 13.2 mph and the rate of the wind is 3.2 mph.