To find the surface area of a cylinder in terms of its radius [tex]\( r \)[/tex] only, given that the cylinder has a volume of 10 cubic units, we can follow these steps:
1. Volume of a Cylinder:
The volume [tex]\( V \)[/tex] of a cylinder is given by the formula:
[tex]\[
V = \pi r^2 h
\][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height. We know the volume is 10 cubic units, so:
[tex]\[
\pi r^2 h = 10
\][/tex]
2. Solve for the Height [tex]\( h \)[/tex]:
To express the height [tex]\( h \)[/tex] in terms of [tex]\( r \)[/tex], we solve the volume equation for [tex]\( h \)[/tex]:
[tex]\[
h = \frac{10}{\pi r^2}
\][/tex]
3. Surface Area of a Cylinder:
The surface area [tex]\( A \)[/tex] of a cylinder is given by the formula:
[tex]\[
A = 2\pi r^2 + 2\pi rh
\][/tex]
The surface area includes the area of the two circular ends [tex]\( (2\pi r^2) \)[/tex] and the rectangular side surface area [tex]\( (2\pi rh) \)[/tex].
4. Substitute [tex]\( h \)[/tex] into Surface Area Formula:
Substitute the expression for [tex]\( h \)[/tex] from step 2 into the surface area formula:
[tex]\[
A = 2\pi r^2 + 2\pi r \left( \frac{10}{\pi r^2} \right)
\][/tex]
Simplify the expression:
[tex]\[
A = 2\pi r^2 + \frac{20}{r}
\][/tex]
5. Simplify Further (if necessary):
The surface area expression can also be written in a more compact form by combining terms:
[tex]\[
A = 2\left( \pi r^3 + \frac{10}{r} \right)
\][/tex]
Therefore, the expression for the surface area [tex]\( A \)[/tex] of the cylinder in terms of the radius [tex]\( r \)[/tex] only is:
[tex]\[
A = 2\pi r^2 + \frac{20}{r}
\][/tex]
or equivalently:
[tex]\[
A = 2\left( \pi r^3 + \frac{10}{r} \right)
\][/tex]
