Answer:
Height or x is 4.50 feet.
Step-by-step explanation:
Surface area is given as : [tex]2\pi rh+2\pi r^{2}[/tex]
Volume is given as : [tex]\pi r^{2} h[/tex]
The value of the surface area of the cylinder is equal to the value of the volume of the cylinder. This means:
[tex]2\pi rh+2\pi r^{2}[/tex] = [tex]\pi r^{2} h[/tex]
Diameter is given as : [tex]7\frac{1}{5} =\frac{36}{5} =7.2[/tex] feet
So, radius = [tex]7.2/2=3.6[/tex] feet
height = h
SA = [tex]2\times3.14\times3.6h+2\times3.14\times(3.6)^{2}[/tex]
= [tex]22.61h+81.39[/tex] feet square
Volume = [tex]3.14\times(3.6)^{2} \times h[/tex]
= 40.70h cubic feet
Now as SA= volume ; we will equal both and solve for h;
[tex]22.61h+81.39=40.70h[/tex]
=> [tex]40.70h-22.61h=81.39[/tex]
=> [tex]18.09h=81.39[/tex]
h = 4.50 feet
Hence, height or x is 4.50 feet.
So, SA = 183.12 square feet
Volume = 183.12 square feet
The volume of a shape is the amount of space in it.
Given that:
[tex]d = 7\frac 15[/tex] -- diameter
[tex]h =x[/tex] --- height
Calculate radius (r)
[tex]r = \frac d2[/tex]
So, we have:
[tex]r = 7\frac 15 \div 2[/tex]
[tex]r = \frac{36}5 \div 2[/tex]
[tex]r = \frac{18}5[/tex]
The volume (V) of a cylinder is:
[tex]V = \pi r^2h[/tex]
So, we have:
[tex]V = \pi (\frac{18}{5})^2 \times x[/tex]
The surface area (A) of a cylinder is;
[tex]A = 2\pi r(r + h)[/tex]
So, we have:
[tex]A = 2\pi \times \frac{18}{5}(\frac{18}{5} + x)[/tex]
The volume and the surface area have the same value.
This means
[tex]2\pi \times \frac{18}{5}(\frac{18}{5} + x) = \pi (\frac{18}{5})^2 \times x[/tex]
Cancel out common terms
[tex]2 \times (\frac{18}{5} + x) = \frac{18}{5} \times x[/tex]
Divide through by 2
[tex]\frac{18}{5} + x = \frac{9}{5} \times x[/tex]
[tex]\frac{18}{5} + x = \frac{9x}{5}[/tex]
Collect like terms
[tex]\frac{18}{5} = \frac{9x}{5} - x[/tex]
Take LCM
[tex]\frac{18}{5} = \frac{9x - 5x}{5}[/tex]
[tex]\frac{18}{5} = \frac{4x}{5}[/tex]
Cancel out common factors
[tex]18} = 4x[/tex]
Divide by 4
[tex]4.5 = x[/tex]
Hence:
[tex]x = 4.5[/tex]
Recall that, the equation of area is:
[tex]A = 2\pi \times \frac{18}{5}(\frac{18}{5} + x)[/tex]
So, we have:
[tex]A = 2\pi \times \frac{18}{5}(\frac{18}{5} + 4.5)[/tex]
Rewrite as:
[tex]A = 2\pi \times \frac{18}{5}(\frac{18}{5} + \frac 92)[/tex]
Take LCM
[tex]A = 2\pi \times \frac{18}{5}(\frac{36 + 45}{10})[/tex]
[tex]A = \pi \times \frac{18}{5}(\frac{36 + 45}{5})[/tex]
[tex]A = \pi \times \frac{18}{5}(\frac{81}{5})[/tex]
[tex]A = \frac{1458}{25}\pi[/tex]
Hence, the volume and the area of the cylinder is: [tex]\frac{1458}{25}\pi[/tex]
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