The area of the cylinder is a function of its height (h) and radius, ([tex]\mathbf{4\sqrt 3}[/tex])
The exact value of h is: [tex]\mathbf{7\sqrt 2- 4\sqrt 3}[/tex]
The given parameters are:
[tex]\mathbf{Area =56\pi\sqrt 6}[/tex]
[tex]\mathbf{r=4\sqrt 3}[/tex]
The surface area of a cylinder is calculated as:
[tex]\mathbf{Area = 2\pi rh + 2\pi r^2}[/tex]
Substitute values for Area
[tex]\mathbf{56\pi\sqrt 6= 2\pi rh + 2\pi r^2}[/tex]
Divide through by pi
[tex]\mathbf{56\sqrt 6= 2 rh + 2r^2}[/tex]
Substitute value for r
[tex]\mathbf{56\sqrt 6= 2 (4\sqrt 3)h + 2(4\sqrt 3)^2}[/tex]
[tex]\mathbf{56\sqrt 6= 8h\sqrt 3 + 2\times 48}[/tex]
[tex]\mathbf{56\sqrt 6= 8h\sqrt 3 + 96}[/tex]
Collect like terms
[tex]\mathbf{8h\sqrt 3 = 56\sqrt 6- 96}[/tex]
Make h the subject
[tex]\mathbf{h = \frac{56\sqrt 6}{8\sqrt 3}- \frac{96}{8\sqrt 3}}[/tex]
[tex]\mathbf{h = 7\sqrt 2- \frac{12}{\sqrt 3}}[/tex]
Rationalize
[tex]\mathbf{h = 7\sqrt 2- \frac{12\sqrt 3}{3}}[/tex]
[tex]\mathbf{h = 7\sqrt 2- 4\sqrt 3}[/tex]
Hence, the exact value of h is: [tex]\mathbf{7\sqrt 2- 4\sqrt 3}[/tex]
Read more about surface areas t:
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