The formula for the Volume of a cone is:

#V = pir^2h/3#

Where:

#V# is the Volume of the cone: #141.3"in"^3# for this problem.

#r# is the radius of the cone: what we are solving for in this problem.

#h# is the height of the cone: #15"in"# for this problem.

Substituting and solving for #r# gives:

#141.3"in"^3 = pi xx r^2 xx (15"in")/3#

#141.3"in"^3 = pi xx r^2 xx 5"in"#

#(141.3"in"^3)/color(red)(5"in") = (pi xx r^2 xx 5"in")/color(red)(5"in")#

#(141.3"in"^(color(red)(cancel(color(black)(3)))2))/color(red)(5color(black)(cancel(color(red)("in")))) = (pi xx r^2 xx color(red)(cancel(color(black)(5"in"))))/cancel(color(red)(5"in"))#

#(141.3"in"^2)/color(red)(5) = pir^2#

#28.26"in"^2 = pir^2#

#(28.26"in"^2)/color(red)(pi) = (pir^2)/color(red)(pi)#

#(28.26"in"^2)/color(red)(pi) = (color(red)(cancel(color(black)(pi)))r^2)/cancel(color(red)(pi))#

#(28.26"in"^2)/color(red)(pi) = r^2#

We can use 3.1416 to estimate #pi# giving:

#(28.26"in"^2)/color(red)(3.1416) = r^2#

#9"in"^2 = r^2# rounded to the nearest inch.

Now, take the square root of each side of the equation to find the radius of the cone while keeping the equation balanced:

#sqrt(9"in"^2) = sqrt(r^2)#

#3"in" = r#

#r = 3"in"#

The radius of the cone rounded to the nearest inch is 3 inches.