Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 1 - Functions

Problem 1.3.3

Chapter 1, Problem 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?

Verified step by step guidance

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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.

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Related Practice

Textbook Question

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

Textbook Question

Theory and Examples

In Exercises 69 and 70, match each equation with its graph. Do not use a graphing device, and give reasons for your answer.

a. y = x⁴

b. y = x⁷

c. y = x¹⁰

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Textbook Question

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

Textbook Question

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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Textbook Question

In Exercises 9–16, determine whether the function is even, odd, or neither.

𝔂 = x² + 1

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Textbook Question

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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