Answer:
Step-by-step explanation:
The bag of sand wave is
W=144lb at x=0
The bag of sand is lifted at a constant rate, i.e the it is not accelerating, so a=0m/s²
Let W be the mass of the sand,
Sand leaks out at a constant rate
dW/dx= C
At a height=18ft, half of the original mass.
i.e x=18ft, M=72lb
We are asked to find the work at 18ft
Work is given as
Work=∫F•dx,
Where F is the force and also the weight of the sand, F=W
The Weight of the bag is a linear function,
dW/dx= C
Then, using variable separation
dW=Cdx
Integrating both sides
∫dW=∫Cdt
W=Cx + B
So, at x=0, W=144
144=B
B=144
W=Cx+144
Also at x=18, W=72lb
72=C×18+144
72-144=18C
-72=18C
C=-4
Then, the weight function becomes
W=-4x+144
W=144-4x
Then applying work formula
Work=∫W•dx
Work=∫(144-4x)dx. x=0 to x=18
Work = 144x-4x²/2. x=0 to x=18
Work=144x-2x² x=0 to x=18
Work=144(18)-2(18²) -0 -0
Work =2592-648
Work =1944 ft lbs
The work done in lifting the sand to 18ft is 1944 ft lbs
