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Zorluk: ZorSurface Area and Volume of 3D Solids

A solid right circular cylinder of radius 5 cm5\text{ cm} and height 12 cm12\text{ cm} has a conical cavity of the same radius and height hollowed out from its top face. What is the total surface area of the remaining solid in cm2\text{cm}^2, expressed as a multiple of π\pi (that is, find the value of KK where the total surface area is Kπ cm2K\pi\text{ cm}^2)?

Cevap: 210 cm²

Cevap

The total surface area of the remaining solid is 210π cm2210\pi\text{ cm}^2, so the required numerical coefficient KK is 210210.
The total surface area consists of three parts: the flat circular base at the bottom (25π cm225\pi\text{ cm}^2), the outer curved surface of the cylinder (120π cm2120\pi\text{ cm}^2), and the newly created inner curved surface of the conical cavity (65π cm265\pi\text{ cm}^2). Adding these together yields 25π+120π+65π=210π cm225\pi + 120\pi + 65\pi = 210\pi\text{ cm}^2, giving the coefficient K=210K = 210.

Adım Adım Çözüm

1
Calculate the slant height (ll) of the conical cavity.
l=52+122=169=13 cml = \sqrt{5^2 + 12^2} = \sqrt{169} = 13\text{ cm}.
The slant height forms the hypotenuse of the right triangle whose leg lengths are the base radius and the vertical height of the cone.
2
Determine the area of all exposed boundary surfaces of the remaining solid.
Flat bottom base area =π(5)2=25π cm2= \pi(5)^2 = 25\pi\text{ cm}^2; Outer curved cylindrical surface area =2π(5)(12)=120π cm2= 2\pi(5)(12) = 120\pi\text{ cm}^2; Inner curved conical surface area =π(5)(13)=65π cm2= \pi(5)(13) = 65\pi\text{ cm}^2.
Hollowing out the cone creates an internal curved boundary while leaving the outer cylindrical boundary and the flat bottom base exposed.
3
Sum the areas of all exposed surfaces to calculate the total surface area.
Total Surface Area =25π+120π+65π=210π cm2= 25\pi + 120\pi + 65\pi = 210\pi\text{ cm}^2.
The total surface area is the sum of all external and internal exposed surface areas.

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