Answer:
The answer to the question is
W = 25 × (Building height) - 30× ( total time taken) + (Building height)²
which is of the form W = Y² + 25·Y - 30·t where Y = Building height and t = total time from the start to the top of the building
Step-by-step explanation:
Firstly let us take the height as = Y
consider the work done in lifting the bucket,
= = (2·Y²/2) = Y² ft·lb
Next to consider the work done in lifting the water alone we have
The weight at the top of the building = 5 lb
Weight at the start of the lift = 30 lb
The work done in lifting the bucket from level x to level x - dx is the product of the weight dx
Therefore the total work in lifting the bucket is
The position is time dependent thus x = Y - Vt ft
The water weight after t seconds is 30 - tl lb, where l = leak rate
where t1l = 25 lb total time =
Therefore dx = -Vdt
Substituting we have vt = Y and lt = 25 lb substituting t at
= [30·t -t²·L·V]ᵇ₀ = -30·b +b²× l×v = b(-30 + blv) but bv = Y and bl = 25
Therefore we have → 25Y- 30b
or W = 25 × (Building height) - 30× ( total time taken) + (Building height)²
which is of the form W = Y² + 25·Y - 30·t
To solve a specific case, let height of the building be 5 ft and the cable weighs 1.5 lb/ft pull rope speed = 1 ft/s
bucket leak rate = 25 lb in 5 seconds = 5 lb/s
To calculate the work done in lifting the cable we have
considering a small length of cable dx located at x below the top has a weight of 1.5 dx, the required work to completely pull the total length of the cable to the top is dW = 1.5 x dx, that is
= [x²/4]⁵₀ = 25/4 ft·lb
For the bucket without water we have
= 5 × 2 = 10 ft·lb
Next we calculate the work done is given by considering a displacement dx of the water from level x = weight of the water at x × dx that is the total work done is given by
as the level of the water at a particular time is dependent on time we have 30 - 5·t lb and the distance to the top of the building = 5 -t substituting, we have
[-30·t + 5·t²/2]⁵₀ = 87.5 ft·lb
Therefore the answer is
Total work done = = 25/4 ft·lb + 10 ft·lb + 87.5 ft·lb = 103.75 ft·lb