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, -h/2`<=` x `<=` h/2, where c is a positive constant. Show that the radius of each end of the barrel is r = R-d, where d = (ch^2)/4.
Given curve `y=R-cx^2` , which is parabola and open down.
Let us write this parabola as
`=>` Parabola is defined if `R-y>=0`
From above equation , we have
`x=0 => y=R` and `y=0=>x=+-sqrt(R/c)`
A barrel with height h and maximum radius R is constructed by rotating about the x-axis the parabola y = R-cx^2, -h/2 x h/2, where c is a positive constant.
`=> ` If `x=h/2` then the value of y will be the radius of the barrel.
Thus the radius of the barrel `r=y=R-(ch^2)/4`
`r=R-d ,` where `d=(ch^2)/4` .