Skip to main content
Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblema 4.5.5a
Capitolo 4, Problema 4.5.5a

Suppose the objective function P= xy is subject to the constraint 10x + y = 100, where x and y are real numbers.


a. Eliminate the variable y from the objective function so that P is expressed as a function of one variable x.

Guida verificata passo dopo passo
1
Start with the constraint equation: 10x + y = 100.
Solve the constraint equation for y in terms of x: y = 100 - 10x.
Substitute the expression for y from the constraint into the objective function P = xy.
This substitution gives P = x(100 - 10x).
Simplify the expression to express P as a function of x: P(x) = 100x - 10x^2.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
1m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Objective Function

An objective function is a mathematical expression that defines a quantity to be maximized or minimized, often subject to certain constraints. In this case, the objective function P = xy represents a product of two variables, x and y, which we aim to optimize under the given constraint.
Video consigliato:
Percorso guidato
06:21
Properties of Functions

Constraints

Constraints are conditions or limitations placed on the variables of an optimization problem. Here, the constraint 10x + y = 100 restricts the values that x and y can take, ensuring that any solution must satisfy this linear equation while optimizing the objective function.
Video consigliato:
10:13
Intro to Applied Optimization: Maximizing Area

Substitution Method

The substitution method involves replacing one variable in an equation with an expression derived from another equation. In this problem, we will use the constraint to express y in terms of x, allowing us to rewrite the objective function P solely in terms of x, simplifying the optimization process.
Video consigliato:
05:21
Finding Limits by Direct Substitution
Pratica correlata
Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. F(x) = x³ - 4x + 100 and G(x) = x³ - 4x - 100 are antiderivatives of the same function.

52
views
Domanda del libro di testo

Folded boxes


a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 5 ft by 8 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way.

319
views
Domanda del libro di testo

Optimal soda can


a. Classical problem Find the radius and height of a cylindrical soda can with a volume of 354 cm³ that minimize the surface area.

292
views
Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. The function f(x) = √x has a local maximum on the interval [0,∞).

234
views
Domanda del libro di testo

Maximizing profit Suppose a tour guide has a bus that holds a maximum of 100 people. Assume his profit (in dollars) for taking people on a city tour is P(n) = n(50 - 0.5n) - 100. (Although P is defined only for positive integers, treat it as a continuous function.)


a. How many people should the guide take on a tour to maximize the profit?

342
views
Domanda del libro di testo

Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.


a. With K = 300 and b = 30, what is lim_t→∞ P(t), the carrying capacity of the population?

333
views