To solve for the positive value of [tex]\( r \)[/tex] in terms of [tex]\( h \)[/tex] and [tex]\( V \)[/tex], follow these steps:
1. The original formula for the volume of a cone is given by:
[tex]\[
V = \frac{1}{3} \pi r^2 h
\][/tex]
2. Start by isolating [tex]\( r^2 \)[/tex] on one side of the equation. First, multiply both sides of the equation by 3 to get rid of the fraction:
[tex]\[
3V = \pi r^2 h
\][/tex]
3. Next, divide both sides of the equation by [tex]\( \pi h \)[/tex] to isolate [tex]\( r^2 \)[/tex]:
[tex]\[
r^2 = \frac{3V}{\pi h}
\][/tex]
4. Finally, take the square root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \sqrt{\frac{3V}{\pi h}}
\][/tex]
This is the positive value of [tex]\( r \)[/tex] in terms of [tex]\( h \)[/tex] and [tex]\( V \)[/tex]:
[tex]\[
r = \sqrt{\frac{3V}{\pi h}}
\][/tex]
So, the rewritten formula for the radius [tex]\( r \)[/tex] of the cone in terms of the volume [tex]\( V \)[/tex] and the height [tex]\( h \)[/tex] is:
[tex]\[
r = \sqrt{\frac{3V}{\pi h}}
\][/tex]
