Question

A culvert is constructed out of large cylindrical shells cast in concrete, as shown in the figure. Using the formula for the volume of a cylinder given on the inside back cover of this book, explain why the volume of the cylindrical shell is V = πR2h − πr2h. Factor to show that V = 2π · average radius · height · thickness Since average radius = Incorrect: Your answer is incorrect. , height = h, and thickness = Incorrect: Your answer is incorrect. , we see the following. V = πR2h − πr2h = 2πh 2 = 2πh(R − r) 2 = 2π · 2 · h · R − r = 2π · average radius · height

Answer 1

A cylindrical shell has a hollow part that is found inside a cylinder. Because the inside part is hollow, you don't account that as part of the total volume. If you look at the picture attached, the hollow part is the red area. We only account for the yellow area because that is the volume of concrete. Hence, the volume is difference of the outer volume to the inner volume.

Volume of Cylindrical Shell = Volume of Outside cylinder - Volume of inside cylinder
Since the volume of a cylinder is πr²h,
Volume of Cylindrical Shell = πR²h - πr²h