Question

You are the building manager for a cylindrical skyscraper with a slanted roof. You must contract a window cleaner fwho works for $35 an hour and can clean 1 square meter of glass per minute. If you know the building is 120m tall at its highest point and 100m tall at its lowest point, and 20m wide, how much should you budget for the contract? Use a line integral of a scalar function to solve. Assume there are no windows to be cleaned on the roof.

Answer 1

Let us think that the base of the cylinder is given by circle in the parametric form => Cir(x,y)

x² + y² = r² (Width or Diameter = 20, hence radiaus r = 10)

This circle is centred at (0,0,0).

or in the parametric form

x = r cos t, y = r sin t

=> dx/dt = - r sin t and dy/dt = r cost t

hence, absolute value of d[Cir(x,y)] / dt = r

now the, slanted slope has a height of 20 m (highest height - lowest height)

Also this slanted slope has base of 20 m. Hence, this slope is slanted with the x,y plane at 45 degree.

Hence, this plane goes through points (-10,0,120), (10,0,100) and (0,10,110).

Hence, using the vector cross product of any of the vectors using these three points, we get the equation of the surface of the roof be

x + z = 110 (verifiable using any of the previous points)

z = 110 - x = 110 - r cos t

hence, the surface of the cylinder =

=

hence, Time taken to clean the total area = hours

Hence, the total budget should be = $