🤔 Find out what you don't know with free Quizzes 🤔Start Quiz Now!

Like

Report

WZ
Numerade Educator

Like

Report

Problem 71 Hard Difficulty

Some of the pioneers of calculus, such as Kepler and Newton, were inspired by the problem of finding the
volumes of wine barrels. (In fact Kepler published a book Stereometria doliorum in 1615 devoted to methods for finding the volumes of barrels.) They often approximated the shape of the sides by parabolas.
(a) A barrel with height $ h $ and maximum radius $ R $ is constructed by rotating about the x-axis the parabola $ y = R - cx^2 $, $ \frac{-h}{2} \le x \le \frac{h}{2} $, where c is a positive constant. Show that the radius of each end of the barrel is $ r = R - d $, where $ d = \frac{ch^2}{4} $.
(b) Show that the volume enclosed by the barrel is
$$ V = \frac{1}{3} \pi h (2R^2 + r^2 - \frac{2}{5} d^2) $$

Answer

a) $R-d=r$
b) $V=\frac{1}{3} \pi h\left(2 R^{2}+r^{2}-\frac{2}{5} d^{2}\right)$

Discussion

You must be signed in to discuss.

Video Transcript

The problem is for pate a barrel with hat age and the maximum writers are it's constructed by rotating. About is the X axis the proble? Why you could are minus e x squire access betweennegative h over to Andi H over too where c is a positive constant showed that the writers of each end of the barrel is small eyes. They're goingto pepto arm honesty. Why ar d is he called to say it's quite over four The four part eh? What we need to do is just parking and over to toe axe we have why his legal Chu our Linus sede times It's over two square This is because you hard my honesty for part two showed that the volume in closed the last barrel ISS the third one third part age comes to our school Gym class are square mons to our father he square So for this problem have the culture integral from negative It's over two to hitch over too Hi are minus C square Squire attacks So this is it true to a tamps into girl from zero to H over too. Hi our squire minus you are see ex squire us say square acts to force, do you? Thanks that this is equal to you. Pie. Eight times I score over two minus R C and squire who over twelve plus say square page two or over. Why sixty now? The comm pyre. This result on DH This vomiting So first, let's compute Teo Square a small square minus two over five. He's square. This is control to our squire. Us small eyes Go Teo kept o r my honesty squatting and then Linus, welcome to our fall Comes say squire age your force over sixteen, this is Echo two three are square minus two are he's a He's a C H squire off or on class. The squire's so this Stay square to force over sixteen minus two Over five. Say square, pitch your force over sixteen, This is control three. Our squire minus. You are say H square, our or us three or five city square to force over sixteen. Then my third of pie age times. This one says this is he caught you. Hi, Paige Times. Ah, Squire Minus you have. Here's this to Han se scoring. Plus, love comes sixteen. You know this is going to force. So this is the same as this result. So we have V is equal to one third page terms to our square past a squire menace. Two or five comes this way.