교과서 질문
Use graphing software to graph the functions specified in Exercises 31–36.
Select a viewing window that reveals the key features of the function.
Graph the upper branch of the hyperbola y² − 16x² = 1.
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
1장, 문제 1.3.3
[Technology Exercise]
You want to make an 80° angle by marking an arc on the perimeter of a 12-in.-diameter disk and drawing lines from the ends of the arc to the disk’s center. To the nearest tenth of an inch, how long should the arc be?
검증된 단계별 안내1
First, understand that the arc length of a circle is a portion of the circle's circumference. The formula for the arc length (L) is given by L = θ/360° * C, where θ is the angle in degrees and C is the circumference of the circle.
Calculate the circumference of the circle using the formula C = π * d, where d is the diameter of the circle. Here, the diameter is 12 inches.
Substitute the given values into the circumference formula: C = π * 12 inches.
Now, substitute the values of θ = 80° and the calculated circumference into the arc length formula: L = 80/360 * C.
Simplify the expression to find the arc length L, ensuring to round the final result to the nearest tenth of an inch.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
1m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Arc Length
Arc length is the distance along the curved line of a circle's circumference. It can be calculated using the formula: Arc Length = (θ/360) × 2πr, where θ is the angle in degrees and r is the radius of the circle. In this case, the radius is half the diameter, which is 6 inches.
추천 영상:
11:35
Example 6: Shortest Rope
Circle Geometry
Circle geometry involves the properties and relationships of circles, including their angles, radii, and chords. Understanding how angles relate to arcs is crucial, as the angle subtended at the center of the circle directly influences the length of the arc. For an 80° angle, this relationship is key to finding the correct arc length.
추천 영상:
06:35Changing Geometries
Radians vs. Degrees
Radians and degrees are two units for measuring angles. While degrees are more commonly used in everyday contexts, radians are often preferred in calculus and geometry due to their direct relationship with arc length and circle properties. To convert degrees to radians, use the formula: radians = degrees × (π/180). This conversion may be necessary for certain calculations.
추천 영상:
가이드 코스
5:04Converting between Degrees & Radians
관련 실천
교과서 질문
In Exercises 35 and 36, find the (a) domain and (b) range.
𝔂 = { -x - 2, -2 ≤ x ≤ - 1
{ x, -1 < x ≤ 1
{ -x + 2, 1 < x ≤ 2
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교과서 질문
Shifting Graphs
Exercises 27–36 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.
y = (1/2)(x + 1) + 5 Down 5, right 1
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교과서 질문
Increasing and Decreasing Functions
Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
y = x³/8
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교과서 질문
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x² + 1
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교과서 질문
Functions and Graphs
Express the area and circumference of a circle as functions of the circle’s radius. Then express the area as a function of the circumference.
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