Calculate the work against gravity required to build a tower of height 35 ft in the shape of a right circular cone with base of radius 7 ft out of brick. Assume that brick has density .
___________ ft-lbs.
The work required to raise a mass $\displaystyle m$ to a height $\displaystyle h$ is $\displaystyle w=mgh$.
The radius of the cone at a height $\displaystyle 0\le h\le 35$ feet is:
$\displaystyle r=-\frac{h}{5}+7$
The volume of a slice of the cone perpendicular to the axis at a height $\displaystyle h$ and of thickness $\displaystyle \Delta h$ is:
$\displaystyle V_{\Delta h}(h)=\pi r^2 \Delta h=\pi (7-(h/5))^2 \Delta h$
Now the work can be seen to be:
$\displaystyle \int_{h=0}^{35} \pi r^2 \rho\, g\, h \; dh=\int_{h=0}^{35}\pi (7-(h/5))^2 \rho\, g\, h \; dh$
CB