교과서 질문
29–32. Intersection points Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.
r = 2 cos θ and r = 1 + cos θ
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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
12장, 문제 12.3.87
85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)
A circular corral of unit radius is enclosed by a fence. A goat is outside the corral and tied to the fence with a rope of length 0≤a ≤ π (see figure). What is the area of the region (outside the corral) that the goat can reach?
검증된 단계별 안내1
Identify the problem setup: A goat is tied outside a circular corral of radius 1 with a rope of length \(a\), where \(0 \leq a \leq \pi\). The goat can move along the fence and also away from it, but only up to the length of the rope.
Visualize the reachable area: The goat's reachable region outside the corral is formed by the path along the fence (an arc of length \(a\)) plus the area the goat can graze around the point where the rope ends, which forms a circular segment outside the corral.
Express the length of the arc along the fence: Since the corral has radius 1, the length of the arc the goat can move along is \(a\), which corresponds to an angle \(\theta = a\) radians on the unit circle.
Calculate the area accessible to the goat: The total reachable area outside the corral is the sum of the area of the circular segment formed by the rope length \(a\) and the area of the circular sector of radius \(a\) that the goat can graze beyond the fence. This can be expressed as the area of a circular sector of radius \(a\) minus the area of the triangular portion under the chord, plus the area of the arc segment along the fence.
Formulate the area mathematically: Use the formula for the area of a circular sector \(\frac{1}{2} a^2 \theta\) and the area of the triangular portion \(\frac{1}{2} a^2 \sin(\theta)\) to write the grazing area as \(\frac{1}{2} a^2 (\theta - \sin(\theta))\), where \(\theta = a\). Combine this with the arc length \(a\) along the fence to get the total reachable area.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Geometry of Circles and Arcs
Understanding the properties of circles, including radius, circumference, and arc length, is essential. The goat is tied to a point on the circular fence, and the rope length determines the arc along which the goat can move. Calculating the reachable area involves analyzing segments and arcs of the circle.
추천 영상:
가이드 코스
06:29
Arc Length of Parametric Curves
Area Calculation Using Integration
To find the area accessible to the goat, one must set up and evaluate integrals that represent the region outside the corral but within the rope's reach. This involves integrating over circular segments or sectors, considering the constraints imposed by the fence and rope length.
추천 영상:
가이드 코스
07:59Estimating the Area Under a Curve Using Left Endpoints
Parametric and Polar Coordinates
Using parametric or polar coordinates simplifies the description of points on and around the circular corral. These coordinate systems help express the boundary conditions and limits of integration more naturally when dealing with circular shapes and arcs.
추천 영상:
05:32Intro to Polar Coordinates
관련 실천
교과서 질문
31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.
(-4, 4√3)
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교과서 질문
Subtle symmetry Without using a graphing utility, determine the symmetries (if any) of the curve r=4-sin (θ/2)
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교과서 질문
9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.
(-1, -π/3)
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교과서 질문
49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.
(x - 1)² + y² = 1
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교과서 질문
11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.
r = 2θ; (π/2, π/4)
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