A right circular cylinder with a height of 10 cm and a surface area of S cm2 has a radius given by r(S)=1/2(√100+2S/π −10).Find lim S→0^+ r(S) and interpret your result.

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

문제 79

2장, 문제 79

A right circular cylinder with a height of 10 cm and a surface area of S cm² has a radius given by r(S)=1/2(√100+2S/π −10).
Find lim S→0^+ r(S) and interpret your result.

검증된 단계별 안내

Identify the function for the radius: \( r(S) = \frac{1}{2}(\sqrt{100 + \frac{2S}{\pi}} - 10) \).

Understand that you need to find \( \lim_{S \to 0^+} r(S) \).

Substitute \( S = 0 \) into the function: \( r(0) = \frac{1}{2}(\sqrt{100 + \frac{2 \cdot 0}{\pi}} - 10) \).

Simplify the expression: \( r(0) = \frac{1}{2}(\sqrt{100} - 10) \).

Calculate the limit by evaluating the simplified expression.

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주요 개념

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

Limits

In calculus, a limit describes the behavior of a function as its input approaches a certain value. It is essential for understanding continuity, derivatives, and integrals. In this context, we are interested in the limit of the radius function r(S) as S approaches 0 from the positive side, which helps us determine the behavior of the radius when the surface area is minimal.

Surface Area of a Cylinder

The surface area of a right circular cylinder is calculated using the formula S = 2πr(h + r), where r is the radius and h is the height. This concept is crucial for understanding how the radius r(S) is derived from the surface area S, and it provides context for the relationship between the dimensions of the cylinder and its surface area.

Function Interpretation

Interpreting the result of a limit involves understanding what the limit signifies in the context of the problem. In this case, finding lim S→0^+ r(S) will reveal the radius of the cylinder as the surface area approaches zero, which can provide insights into the geometric implications of a cylinder with minimal surface area, such as its shape and dimensions.