Question

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Answer 1

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:

`V=\pi r^2h`

Formula for Surface area of a cylinder:

`SA=2(\pi rh+\pi r^2)`

Since the values for both surface area and volume are equal, equate both formula:

`\begin{gathered} SA=V \\ \\ 2(\pi rh+\pi r^2)=\pi r^2h \end{gathered}`

Let's simplify the equation.

Divide through by πr:

`\begin{gathered} (2\pi rh)/(\pi r)+(2\pi r^2)/(\pi r)=(\pi r^2h)/(\pi r) \\ \\ 2h+2r=rh \end{gathered}`

Write the equation for h.

`\begin{gathered} 2h-rh=-2r \\ \text{Factor h out of 2h and rh} \\ h(2-r)=-2r \\ \\ \text{Divide both sides by 2-r:} \\ (h(2-r))/((2-r))=-(2r)/(2-r) \\ \\ h=(-2r)/(2-r) \end{gathered}`

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:

`\begin{gathered} h=-(2(3.6))/(2-3.6) \\ \\ h=-(7.2)/(-1.6) \\ \\ h=4.5\text{ ft} \end{gathered}`

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: