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Chapter 4: Problem 1

Consider the following problem: Find two numbers whose sum is 23 and whose product is a maximum. (a) Make a table of values, like the following one, so that the sum of the numbers in the first two columns is always \(23 .\) On the basis of the evidence in your table, estimate the answer to the problem. (b) Use calculus to solve the problem and compare with your answer to part (a).

Short Answer

Expert verified
The numbers are 11.5 and 11.5.

Step by step solution

01

Create a Table of Values

To find two numbers whose sum is 23 and whose product is maximized, list possible pairs of numbers (x and y) such that their sum is 23. Fill in a table with columns for `Number 1 (x)`, `Number 2 (y)`, and `Product (xy)`. For example: | Number 1 (x) | Number 2 (y) | Product (xy) | |--------------|--------------|--------------| | 1 | 22 | 1 * 22 = 22 | | 2 | 21 | 2 * 21 = 42 | | 3 | 20 | 3 * 20 = 60 | | ... | ... | ... | | 11.5 | 11.5 | 11.5 * 11.5 = 132.25 | By completing this table, you can observe that the product increases, reaches a maximum at the pair (11.5, 11.5), and then begins to decrease again.
02

Set Up the Function for the Product

Let one number be represented by \(x\), then the second number is \(23 - x\). The function for the product \(P(x)\) is given by:\[P(x) = x(23 - x) = 23x - x^2\]We need to find the maximum value of this function.
03

Differentiate the Product Function

To find the critical points, differentiate the product function:\[P'(x) = \frac{d}{dx}(23x - x^2) = 23 - 2x\]Set the derivative equal to zero to find the critical points:\[23 - 2x = 0\]
04

Solve for Critical Points

Solve the equation from the previous step:\[23 = 2x \x = 11.5\]This critical point \(x = 11.5\) is a candidate for the maximum product.
05

Verify the Maximum

To confirm that \(x = 11.5\) gives a maximum product, check the second derivative:\[P''(x) = \frac{d}{dx}(23 - 2x) = -2\]Since \(P''(x) < 0\), the function has a local maximum at \(x = 11.5\).
06

Conclusion

The two numbers you were seeking, whose sum is 23 and product is a maximum, are both \(11.5\). Comparing with the table from Step 1, this corresponds with the observed maximum product.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Critical Points
In the context of optimization problems, critical points play a crucial role. They are the points in the domain of a function at which the derivative is zero or undefined, indicating potential locations for maximum or minimum values. In the exercise given, we are dealing with the function for the product of two numbers, \( P(x) = 23x - x^2 \). To find potential maximum values, we begin by differentiating this function to obtain the derivative, \( P'(x) = 23 - 2x \). By setting this derivative equal to zero, \( P'(x) = 0 \), we can solve for \( x \) to find our critical point. Here this results in \( x = 11.5 \), a key value where the product of two numbers might achieve its highest possible value.
  • Critical points can indicate where functions reach their maximum or minimum values.
  • Finding them usually involves differentiating and solving \( f'(x) = 0 \).
Second Derivative Test
The second derivative test is a method used to determine the nature of a critical point. Specifically, it can tell us whether a given critical point is a local maximum, local minimum, or neither. In this exercise, once we have our critical point at \( x = 11.5 \), we need to apply the second derivative test to confirm that this indeed provides a maximum product. For this, we calculate the second derivative of the product function: \( P''(x) = -2 \).Since \( P''(x) \) is less than zero, it indicates that the function \( P(x) \) has a concave down shape at that point, and therefore, \( x = 11.5 \) gives a local maximum.
  • Use the second derivative to classify critical points.
  • If \( f''(x) < 0 \), the point is a maximum; if \( f''(x) > 0 \), it is a minimum.
Product Maximization
The goal of product maximization in this problem is to find two numbers whose product is as large as possible under a given constraint. We need to identify an operation or function that represents the relationship, such as the product of two numbers here.In a formal mathematical approach, we express the product \( P \) as a function of \( x \): \[ P(x) = x(23-x) = 23x - x^2 \]This function captures how the product of two numbers changes as \( x \) varies, while their sum remains constant at 23. By identifying the point at which this function reaches its peak, we achieve the maximum product, harnessing both critical points and the derivative test for verification.
  • Represent the product as a mathematical function.
  • Find maximum values using calculus techniques like differentiation.
Sum Constraint
A sum constraint restricts the values of our variables--here the two numbers whose product we aim to maximize. The constraint dictates that these numbers always add up to a certain constant. In our example, the sum of the numbers \( x \) and \( y \) must always be 23, giving us the equation: \[ x + y = 23 \]This constraint allows us to express one variable in terms of the other, simplifying our problem significantly. By substituting \( y = 23 - x \) into our product function, we reduce the number of variables and focus on finding critical points of a single-variable function. Taking advantage of constraints can simplify optimization problems and focus on locating the extreme values efficiently.
  • Crucial for simplifying an optimization problem to a single equation.
  • Allows manipulation of equations to isolate a single variable.

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Most popular questions from this chapter

Coulomb's Law states that the force of attraction between two charged particles is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The figure shows particles with charge 1 located at positions 0 and 2 on a coordinate line and a particle with charge \(-1\) at a position \(x\) between them. It follows from Coulomb's Law that the net force acting on the middle particle is $$F(x)=-\frac{k}{x^{2}}+\frac{k}{(x-2)^{2}} \quad 0< x <2$$ where \(k\) is a positive constant. Sketch the graph of the net force function. What does the graph say about the force?

An object with weight \(W\) is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle \(\theta\) with the plane, then the magnitude of the force is $$F=\frac{\mu W}{\mu \sin \theta+\cos \theta}$$ where \(\mu\) is a positive constant called the coefficient of friction and where 0\(\leqslant \theta \leqslant \pi / 2 .\) Show that \(F\) is minimized when \(\tan \theta=\mu\)

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f^{\prime}\) and \(f^{\prime \prime}\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. $$f(x)=e^{x}-0.186 x^{4}$$

For what values of the numbers \(a\) and \(b\) does the function $$f(x)=a x e^{b x^{2}}$$ have the maximum value \(f(2)=1 ?\)

A high-speed bullet train accelerates and decelerates at the rate of 4 \(\mathrm{ft} / \mathrm{s}^{2}\) . Its maximum cruising speed is 90 \(\mathrm{mi} / \mathrm{h}\) . (a) What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at that speed for 15 minutes? (b) Suppose that the train starts from rest and must come to a complete stop in 15 minutes. What is the maximum distance it can travel under these conditions? (c) Find the minimum time that the train takes to travel between two consecutive stations that are 45 miles apart. (d) The trip from one station to the next takes 37.5 minutes. How far apart are the stations?
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