1) the area of the "side" of the cylinder is A1= pi (4 in).
2) the total area of the circular ends of the cyl. is A2 = 2 pi (2 in)^2 (since the radius of the cyl. is 2 in).
The desired total surface area is A = A1 + A2. Keep "pi;" do not substitute a numerical value for "pi."
Answer:
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<span>The goal in simplifying numbers
is to reduce it that can be best seen in the eye. Also in
converting units, you must take note of the units given in the problem. If you
don’t, you would get a lot of errors. Length is the measure of distance and can
be best represented by either in SI units or English units. Example of an SI unit is meter, m, and English
unit is feet, ft. You are given 3 inches and 2 feet. You can convert any of the
numbers into a number that would feel comfortable to you. I will take 3 inches.
Note that there are 12 inches in 1 feet. So divide 3 by 12 and you will get ¼ feet.
Then divide 2 feet to ¼ feet and you will get 8.</span>
The sector (shaded segment + triangle) makes up 1/3 of the circle (which is evident from the fact that the labeled arc measures 120° and a full circle measures 360°). The circle has radius 96 cm, so its total area is π (96 cm)² = 9216π cm². The area of the sector is then 1/3 • 9216π cm² = 3072π cm².
The triangle is isosceles since two of its legs coincide with the radius of the circle, and the angle between these sides measures 120°, same as the arc it subtends. If b is the length of the third side in the triangle, then by the law of cosines
b² = 2 • (96 cm)² - 2 (96 cm)² cos(120°) ⇒ b = 96√3 cm
Call b the base of this triangle.
The vertex angle is 120°, so the other two angles have measure θ such that
120° + 2θ = 180°
since the interior angles of any triangle sum to 180°. Solve for θ :
2θ = 60°
θ = 30°
Draw an altitude for the triangle that connects the vertex to the base. This cuts the triangle into two smaller right triangles. Let h be the height of all these triangles. Using some trig, we find
tan(30°) = h / (b/2) ⇒ h = 48 cm
Then the area of the triangle is
1/2 bh = 1/2 • (96√3 cm) • (48 cm) = 2304√3 cm²
and the area of the shaded segment is the difference between the area of the sector and the area of the triangle:
3072π cm² - 2304√3 cm² ≈ 5660.3 cm²
