Find the surface area. Enter the exact answer in simplest form.

Answer:

  3024π square inches

Step-by-step explanation:

The surface area is the sum of ...

The relevant area formulas are ...

  A = πrs . . . . . lateral area of cone with radius r and slant height s

  A = 2πrh . . . . lateral area of cylinder with radius r and height h

  A = πr² . . . . . area of a circle of radius r

The radius is half the diameter, so is (36 in)/2 = 18 in. When we add up these areas, we can factor out a common factor to simplify the expression a bit.

  A = πrs +2πrh +πr²

  A = πr(s +2h +r) = π(18 in)(30 in +2(60 in) +18 in) = π(18)(168) in²

  A = 3024π in² . . . . surface area

Check the picture below.

so, to get the area of the object, we'll use the lateral area of a cone with a height of 30 and a radius of 18, and the lateral area of a cylinder with height of 60 and radius of 18, and sum those two up.

[tex]\stackrel{\textit{lateral area of a cone}}{\pi r\sqrt{r^2+h^2}}\implies \pi 18\sqrt{18^2+30^2}\implies 108\pi \sqrt{34} \\\\\\ \stackrel{\textit{lateral area of a cylinder}}{2\pi rh\implies 2\pi (18)(60)}\implies 2160\pi[/tex]

[tex]\stackrel{\textit{area of the bottom circle of the cylinder}}{\pi r^2\implies \pi 18^2\implies 324\pi } \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{total area of the shape}}{108\pi \sqrt{34}+2160\pi +324\pi ~~\approx~~9782.1115~~\implies ~~\stackrel{\textit{rounded up}}{\boxed{9782~~in^2}}}[/tex]12

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