A farmer wishes to paint the side of a cylindrical grain silo of height 80 feet and diameter 32 feet. If the paint is to be applied in a coat 1/8 inch thick, use differentials to approximate the volume of paint that the farmer needs to buy. Calculate the actual error in the approximation; what is the relative error when compared to actual volume of paint required?Thank You!
take a ring of differential thickness dr and of radius r from axis.integrate it from inner to outer radius. it is area. now multiply it by height i.e. 80ft to get the volume. i didn't get ,what approximation?
The silo itself is a cylinder of radius 16 ft and height 80 ft, whereas when painted the radius is increased to [16 + 1/(8*12)] ft. The volume Vo of the paint required is therefore given exactly by the difference in the volumes of these two cylinders Vo = 80*π*(16 + 1/96)? - 80*π*16? = 83.803 cu ft Alternatively, we may calculate an approximation Va to the volume of paint required by taking the product of the surface area of the silo and the thickness of the paint ie Va = 80*2*π*16*(1/96) = 83.775 cu ft which is in error by 0.028 cu ft, or 0.03% of the volume of paint required. This is of course totally negligible in terms of the uncertainties introduced by other factors, principally that of laying down a uniform coat of paint to that thickness and consistency, not to mention the wastage of paint during its application. .
