From the figure, we can deduce the following:
Diameter of the cylinder = 7⅕ ft.
Height of the cylinder = x ft
Here, the surface area of the cylinder is equal to the value of the volume of the cylinder.
Let's solve for x.
Formula for Volume of a cylinder:
[tex]V=\pi r^2h[/tex]Formula for Surface area of a cylinder:
[tex]SA=2(\pi rh+\pi r^2)[/tex]Since the values for both surface area and volume are equal, equate both formula:
[tex]\begin{gathered} SA=V \\ \\ 2(\pi rh+\pi r^2)=\pi r^2h \end{gathered}[/tex]Let's simplify the equation.
Divide through by πr:
[tex]\begin{gathered} \frac{2\pi rh}{\pi r}+\frac{2\pi r^2}{\pi r}=\frac{\pi r^2h}{\pi r} \\ \\ 2h+2r=rh \end{gathered}[/tex]Write the equation for h.
[tex]\begin{gathered} 2h-rh=-2r \\ \text{Factor h out of 2h and rh} \\ h(2-r)=-2r \\ \\ \text{Divide both sides by 2-r:} \\ \frac{h(2-r)}{(2-r)}=-\frac{2r}{2-r} \\ \\ h=\frac{-2r}{2-r} \end{gathered}[/tex]Where:
h is the height
r is the radius.
To find the radius, we have:
radius = diameter/2 = 7.2/2 = 3.6 ft
Now, to find the height, substitute 3.6 for r and evaluate:
[tex]\begin{gathered} h=-\frac{2(3.6)}{2-3.6} \\ \\ h=-\frac{7.2}{-1.6} \\ \\ h=4.5\text{ ft} \end{gathered}[/tex]Therefore, the height of the cylinder, is 4.5 ft.
x = 4.5
Let's find the surface area:
Where:
r = 3.6 ft
h = 4.5 ft
We have:
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