Landon can climb a certain hill at a rate that is 3.5 mph slower than his rate coming down the hill. If it take him 4 hours to climb the hill and 135 minutes to come down the hill. what is his rate coming down.

coming down rate --- x mph

going up rate -----< x-3.5

time going up = hill/(x - 3.5)
hill/(x - 3.5) = 4
hill = 4(x - 3.5)

time going down = hill/x
hill/x = 135/60 = 9/4
hill = (9/4)x

4(x - 3.5) = (9/4)x
16(x-3.5) = 9x
16x - 56 = 9x
7x = 56
x = 8

rate going down is 8 mph
(rate going up is 4.5 mph)

check:
length of hill = (9/4)(8) = 18 miles

time to come down = 18/8 = 2.25 hrs = 135 minutes
time to go up = 18/(8-3.5) = 4 hrs
checks out!

To find Landon's rate coming down the hill, we need to convert the given information into a solvable equation.

1. Let's start by converting the time it took for Landon to come down the hill into hours. We are told that it took him 135 minutes, which is 2 hours and 15 minutes. This conversion can be done by dividing the number of minutes by 60. So, 135 minutes ÷ 60 = 2.25 hours.

2. Next, let's assign a variable to Landon's rate coming down the hill. Let's call it "x" (in mph).

3. We are also told that Landon's rate going up the hill is 3.5 mph slower than his rate coming down the hill. So, his rate going up is (x - 3.5) mph.

4. To find Landon's speed going down the hill, we need to use the formula: Speed = Distance/Time. In this case, we can assume that the distance going up the hill is the same as the distance coming down the hill. So, we will equate the distance and time for both scenarios.

5. Going up the hill: Distance = Rate × Time. Let's substitute the values and solve for the distance: Distance = (x - 3.5) mph × 4 hours = 4(x - 3.5) miles.

6. Coming down the hill: Distance = Rate × Time. Substituting the values and solving for the distance, we get: Distance = x mph × 2.25 hours = 2.25x miles.

7. Since the distance is the same in both scenarios, we can set the two equations equal to each other: 4(x - 3.5) = 2.25x.

8. Simplifying the equation: 4x - 14 = 2.25x.

9. Let's solve for x by moving all the terms with x to one side of the equation: 4x - 2.25x = 14.

10. Combining like terms: 1.75x = 14.

11. Finally, solve for x by dividing both sides of the equation by 1.75: x = 14 ÷ 1.75 = 8.

Therefore, Landon's rate coming down the hill is 8 mph.