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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.5.5a
Chapter 4, Problem 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.

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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.

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Key Concepts

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

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.
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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.
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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.
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Related Practice
Textbook Question

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.

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Textbook Question

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.

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Textbook Question

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.

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Textbook Question

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,∞).

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Textbook Question

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?

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Textbook Question

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?

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