Math Help Forum: Line Integrals

Solve the following using line Integrals:

1) A 160-lb man carries a 25-lb can of paint up a helical staircase that encircles a silo with a radius of 20ft. If the silo is 90ft high and the man makes exactly three complete revolutions, how much work is done by the man against gravity in climbing to the top?

2) If there is a hole in the can of paint in the above problem and 9-lb of paint is leaking steadily out of the can during the man's accent how much work is being done?

I know this can be done using physics but it has to be done using Line Integrals.

Originally Posted by algebrapro18

Solve the following using line Integrals:

1) A 160-lb man carries a 25-lb can of paint up a helical staircase that encircles a silo with a radius of 20ft. If the silo is 90ft high and the man makes exactly three complete revolutions, how much work is done by the man against gravity in climbing to the top?

2) If there is a hole in the can of paint in the above problem and 9-lb of paint is leaking steadily out of the can during the man's accent how much work is being done?

I know this can be done using physics but it has to be done using Line Integrals.

Have you tried to determine the parametric equation for the helix?

The helix is defined parametrically as:



Now, we need to determine a vector field that represents to force exerted by the farmer in lifting his own weight plus the weight of the can of paint. I'll work on that part now, but I'll see if you can figure that out.

Once we get the vector field, we can find the work done:



where



is the force vector field, and



is the vector that represents the path of the field (in this case, the path around the helix)

Note that the Force vector field will be different for both parts. In part A, the weight doesn't change as it follows the path. In part B, the weight decreases at a constant rate, and that is something that will need to be taken into consideration when finding the force field.

Does this make sense? Ask questions if there is something you don't understand.

--Chris

I can't figure out how to get F(x,y,z) though. I can do everything else I just need to find the function.

parameterization of the helix is

x = 20cos2pi(t)

y= 20sin2pi(t)

z= 30(t)

t goes from zero to 3 since we are making three turns

so r(t)=20cos2pi(t)i + 20sin2pi(t)j + 30(t)k

r'(t)= -40pi(sin2pi(t))i +40pi(cos2pi(t)j+30k



F(x,y,z)=0i + 0j + 185j

so F(r(t))= 0i + 0j + 185j

the dot product <0 , 0 , 185> * <-40pi(sin2pi(t)) , 40pi(cos2pi(t), 30>

0+0+185(30)=5550

now just integrate


int(from t=0 to t=3) 5550 dt =16,650 ft lbs


for the second part F(x,y,z) will become 0i + 0y + 185 - (90t/30)j

taking the dot product as above and then integrating you get

16,245 ft lbs for the part where the paint is leaking at a constant rate