A farmer wishes to paint the side of a cylindrical grain sil

A farmer wishes to paint the side of a cylindrical grain silo of height 75 feet and diameter 24 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.
ft³

Calculate the actual error in the approximation.
ft³

What is the relative error when compared to actual volume of paint required? (Round your answer to six decimal places.)

Solution

The silo itself is a cylinder of radius 12 ft and height 75 ft,

whereas when painted the radius is increased to [12 + 1/(8*12.75)] ft. < sice {(75-24)/4=12.75} >

The volume Vo of the paint required is therefore given exactly by the difference in the volumes of these two cylinders Vo = 75**( 12+ 1/102)² - 75**12² =55.2920 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 = 75*2**12*(1/102) = 55.4399 cu ft

which is in error by 0.1479 cu ft, or 0.1% of the volume of paint required.