To find the area of the cylinder we need to find its volume first. Remember that the formula for the volume of a cylinder is [tex]V= \pi r^{2} h[/tex]
where:
[tex]V[/tex] is the volume
[tex]r[/tex] is the radius
[tex]h[/tex] is the height
From the question we know that [tex]A=175[/tex] and [tex]h=7[/tex]. Lets replace those values in our volume formula:
[tex]175= \pi r^{2} 7[/tex]
Now we can solve for [tex]r[/tex] to find our radius:
[tex]r^{2} = \frac{175}{7 \pi } [/tex]
[tex]r^{2} = \frac{25}{ \pi } [/tex]
[tex]r= \sqrt{ \frac{25}{ \pi } } [/tex]
[tex]r= \frac{5}{ \sqrt{ \pi } } [/tex]
Now that we know the radius, we can use the formula for the area of a cylinder [tex]A=2 \pi rh+2 \pi r^{2} [/tex]
where:
[tex]A[/tex] is the area
[tex]r[/tex] is the radius
[tex]h[/tex] is the height
We know now that [tex]r= \frac{5}{ \sqrt{ \pi } } [/tex] and [tex]h=7[/tex], so lets replace those values in our area formula:
[tex]A=2 \pi ( \frac{5}{ \sqrt{ \pi } } )(7)+2 \pi ( \frac{5}{ \sqrt{ \pi } })^{2} [/tex]
[tex]A= \frac{70 \pi }{ \sqrt{ \pi } } +50[/tex]
[tex]A=174.07[/tex]
We can conclude that the area of a cylinder that has a volume of 175 cubic units and a height of 7 units is 174.07 square units.
