Domanda del libro di testo
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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TOC.tableOfContents
- 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
Capitolo 1, Problema 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.
Guida verificata passo dopo passo1
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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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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Percorso guidato
06:21
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.
Video consigliato:
Percorso guidato
06:21Properties of Functions
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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04:27Substitution With an Extra Variable
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