Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = cos(x - 3) + 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 1, Problem 1.1.75
A pen in the shape of an isosceles right triangle with legs of length x ft and hypotenuse of length h ft is to be built. If fencing costs \(5/ft for the legs and \)10/ft for the hypotenuse, write the total cost C of construction as a function of h.
Verified step by step guidance1
Start by identifying the cost components: the cost for the legs and the cost for the hypotenuse. The legs cost \$5 per foot, and the hypotenuse costs \$10 per foot.
Since the triangle is isosceles and right-angled, the two legs are equal in length, each being x feet. Therefore, the cost for the legs is 2 * 5 * x = 10x dollars.
The cost for the hypotenuse, which is h feet long, is 10 * h dollars.
The total cost C of construction is the sum of the costs for the legs and the hypotenuse. Therefore, C = 10x + 10h.
To express C as a function of h, use the Pythagorean theorem for the isosceles right triangle: x^2 + x^2 = h^2, which simplifies to 2x^2 = h^2. Solve for x in terms of h: x = sqrt(h^2/2). Substitute this expression for x into the cost function C = 10x + 10h to express C solely in terms of h.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Isosceles Right Triangle Properties
An isosceles right triangle has two equal legs and a right angle between them. The relationship between the legs (x) and the hypotenuse (h) is defined by the Pythagorean theorem, where h = x√2. Understanding this relationship is crucial for expressing the cost function in terms of the hypotenuse.
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Properties of Functions
Cost Function
A cost function represents the total cost associated with producing a certain quantity of goods or services. In this scenario, the cost function C is determined by the lengths of the triangle's sides and their respective costs: $5/ft for the legs and $10/ft for the hypotenuse. This function will be expressed in terms of h, the hypotenuse.
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Substitution in Functions
Substitution is a method used in algebra to replace a variable with another expression. In this problem, we need to express the lengths of the legs in terms of the hypotenuse h. By substituting x = h/√2 into the cost function, we can derive a total cost function solely in terms of h.
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Related Practice
Textbook Question
Graph the functions in Exercises 37–56.
y = (1/x²) − 1
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Textbook Question
Graph the functions in Exercises 13–22. What is the period of each function?
−cos 2πx
Textbook Question
Graphing
In Exercises 69–76, graph each function not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.14–1.17 and applying an appropriate transformation.
y = (1/2x) − 1
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Textbook Question
Functions
In Exercises 1–6, find the domain and range of each function.
F(x) = √(5x + 10)
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Textbook Question
Evaluate sin (5π/12).