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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 6 - Applications of Integration문제 6.3.69b
6장, 문제 6.3.69b

A right circular cylinder with height R and radius R has a volume of VC=πR^3 (height = radius).


b. Find the volume of the hemisphere that is inscribed in the cylinder with the same base as the cylinder. Express the volume in terms of VC.

검증된 단계별 안내
1
Identify the dimensions of the hemisphere inscribed in the cylinder. Since the hemisphere shares the same base as the cylinder, its radius is equal to the radius of the cylinder, which is \(R\).
Recall the formula for the volume of a hemisphere: \(V_H = \frac{2}{3} \pi r^3\). Substitute \(r = R\) to get \(V_H = \frac{2}{3} \pi R^3\).
Recall the volume of the cylinder given as \(V_C = \pi R^3\) (since height = radius = \(R\)).
Express the volume of the hemisphere \(V_H\) in terms of the cylinder volume \(V_C\) by dividing \(V_H\) by \(V_C\): \(\frac{V_H}{V_C} = \frac{\frac{2}{3} \pi R^3}{\pi R^3}\).
Simplify the expression to find \(V_H\) as a fraction of \(V_C\), which will give the volume of the hemisphere in terms of the cylinder volume.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Volume of a Cylinder

The volume of a right circular cylinder is calculated by multiplying the area of its base by its height. For a cylinder with radius R and height R, the volume is V = πR² × R = πR³. This formula provides the reference volume VC used in the problem.
추천 영상:
04:48
Finding Volume Using Disks

Volume of a Hemisphere

A hemisphere is half of a sphere. The volume of a sphere with radius r is (4/3)πr³, so the volume of a hemisphere is half of that, (2/3)πr³. Understanding this formula is essential to find the volume of the hemisphere inscribed in the cylinder.
추천 영상:
04:48
Finding Volume Using Disks

Inscribed Shapes and Radius Relationship

An inscribed hemisphere inside a cylinder with the same base means the hemisphere's radius equals the cylinder's radius R. Recognizing this relationship allows substitution of R into the hemisphere volume formula and expressing the result in terms of VC.
추천 영상:
07:36
Radius of Convergence
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