Problem $3(15$ points$)$

Large trees were submerged from the creation

of a dam are being logged underwater. The

company doing the logging needs to properly

size the cables to be used to anchor the trees to

the bottom of the reservoir to avoid having the

trees become projectiles to the surface.

Approximate each tree as a truncated cone as

shown in the schematic on the right. The

largest tree has been estimated to have a

height $h$ of $115$ ft $,$ a base diameter $d_b$ of $7$ ft

and a diameter at the top $d_t$ of $2$ ft $.$ The

volume of a truncated cone is

$V=\frac{h}{3}(r_1^2+r_1{r}_2+r_2^2)$

with $h$ being the height of the cone, and $r_1$ and

$r_2$ being the radius at the base and top of the

truncated cone, respectively.

a$)$ Calculate the tension force that should be used to size the cable. Use a specific gravity of $0\mathrm{.}65$ for

the submerged wood.

b$)$ Would you recommend using a safety factor on your result from part $($a$)$ or use this result as is$?$ If

you recommend using a safety factor, what safety factor do you recommend and justify your

answer? If you recommend using the result as is$,$ justify your answer.