An elevator went to the do of a 180m building, stayed for 1 minute, then came down. The total time was 3.5 minutes. If the speed of the elevator when it went up was 1 m/sec less than when it came down, find both speeds (up and down).
To solve this problem, let's assign variables to the unknowns in the question.
Let's call the speed of the elevator when it went up "v" m/sec,
and the speed of the elevator when it came down "v + 1" m/sec.
We'll use the following equation to find the time it took for the elevator to go up:
Time = Distance / Speed
The elevator went up to the top of the 180m building, so the distance is 180m.
Thus, the time it took for the elevator to go up is:
Time going up = 180m / v m/sec
It then stayed at the top for 1 minute. Since 1 minute is equal to 60 seconds, we add that time to the total:
Total time going up = 180m / v m/sec + 60 seconds
Next, we need to find the time it took for the elevator to come down. Since the speed going down is "v + 1" m/sec, we can use the same equation as before:
Time coming down = 180m / (v + 1) m/sec
The total time for the elevator's trip, according to the question, is 3.5 minutes. Since 3.5 minutes is equal to 210 seconds, we can add the time going up and the time coming down to equal 210 seconds:
Total time = Total time going up + Time coming down
3.5 minutes = (180m / v m/sec + 60 seconds) + (180m / (v + 1) m/sec)
Now, we have an equation with one variable, "v". Let's solve it:
3.5 minutes = (180m / v m/sec + 60 seconds) + (180m / (v + 1) m/sec)
First, convert 3.5 minutes to seconds:
3.5 minutes = 3.5 minutes * 60 seconds/minute = 210 seconds
Substituting the values, we have:
210 seconds = (180m / v m/sec + 60 seconds) + (180m / (v + 1) m/sec)
To simplify this equation, let's first subtract 60 seconds from both sides:
150 seconds = 180m / v m/sec + 180m / (v + 1) m/sec
Next, we can multiply both sides by v(v + 1) to get rid of the denominators:
150 seconds * v(v + 1) = 180m(v + 1) + 180m*v
Expanding the left side of the equation:
150v(v + 1) seconds = 180m(v + 1) + 180m*v
150v^2 + 150v = 180m(v + 1) + 180m*v
Now, let's simplify the equation further:
150v^2 + 150v = 180mv + 180m + 180mv
150v^2 + 150v = 360mv + 180m
Moving all the terms to one side, we get:
150v^2 + 150v - 360mv - 180m = 0
This is a quadratic equation. Let's solve it by setting it equal to zero and factoring:
150v^2 + 150v - 360mv - 180m = 0
Rearranging the terms:
150v^2 - 360mv + 150v - 180m = 0
Factoring out common terms:
(v(150v - 360m)) + (30(5v - 6m)) = 0
Now, we have two terms: (v(150v - 360m)) and (30(5v - 6m)).
To find the values of "v" that make this equation true, we set each term equal to zero:
v = 0
150v - 360m = 0
5v - 6m = 0
From the second equation, we can solve for v:
150v = 360m
v = 360m / 150
v = 2.4m/sec
From the third equation, we can solve for v:
5v = 6m
v = 6m / 5
v = 1.2m/sec
Therefore, the speed of the elevator going up was 2.4 m/sec, and the speed of the elevator coming down was 1.2 m/sec.