Respuesta :
The surface area of the grain bin is 288π ft²
What is an area?
Area is the amount of space occupied by a two dimensional shape or object.
The diameter of cylinder = 12 ft, radius(r) = 12/2 = 6, and height (h) = 10 ft. Hence:
Surface area of cylinder = 2π(r² + rh) = 2π(6² + 10*6) = 192π ft²
Cone has height of 8 ft, hence:
Surface area of cone = πr[r + √(r² + h²)] = 6π(6 + √(6² + 8²)) = 96π ft²
Surface area of the grain bin = 192π ft² + 96π ft² = 288π ft²
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Surface area of an object is the measure of the total area its surface possess. The surface area of the grain bin is given by: Option C: 216π sq. ft.
How to find the surface area of a composite figure?
Surface area of the composite figure is the area of the exposed surface of the resultant composite figure to its surroundings.
For the considered figure, the grain bin is formed by joining the cone on top of a cylinder.
The exposed surface of grain bin includes of total surface of cone and cylinder - the top of cylinder and bottom of cone as they now are included inside of the bin.
From the statement of the problem, we have:
- Height of cylinder = 10 ft
- Height of the cone = 8 ft
- Diameter of the cylinder = 12 ft
- Since the cone is on the top, assuming its diameter is also of 12 ft.
Then, we get the surface area of the bin as:
S = Surface area(cylinder) + Surface area(cone) - 2(surface area of top of cylinder)
(it is because base of the cone and top of the cylinder are of same area as they are congruent circles).
Surface area of the considered cylinder = [tex]2\pi r(h+r) = 2\pi (\dfrac{d}{2})(h+\dfrac{d}{2}) = 2\pi . (12/2).(10+12/2) =192\pi \: \rm ft^2[/tex]
Surface area of the considered cone =
[tex]\pi r(r+\sqrt{h^2 + r^2}) = \pi \dfrac{d}{2}(\dfrac{d}{2}+\sqrt{h^2 + (\dfrac{d}{2})^2}) = \pi.6(6+\sqrt{8^2+6^2}) = 96\pi \: \rm ft^2[/tex]
Area of the circle which is base of cone and top of cylinder = [tex]\pi r^2 = \pi (d/2)^2 = \pi (12/2)^2 = 36\pi[/tex]
Thus, we get:
Surface area of grain bin = [tex]192\pi + 96\pi - 36\pi - 36\pi = 216\pi[/tex] (in sq. feet).
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