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Chapter 3: Problem 1

Find two positive numbers whose sum is 110 and whose product is a maximum. (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) $$ \begin{array}{|c|c|c|} \hline \begin{array}{c} \text { First } \\ \text { Number } \boldsymbol{x} \end{array} & \begin{array}{c} \text { Second } \\ \text { Number } \end{array} & \ {\text { Product } \boldsymbol{P}} \\ \hline 10 & 110-10 & 10(110-10)=1000 \\ \hline 20 & 110-20 & 20(110-20)=1800 \\ \hline \end{array} $$ (b) Use a graphing utility to generate additional rows of the table. Use the table to estimate the solution. (c) Write the product \(P\) as a function of \(x\). (d) Use a graphing utility to graph the function in part (c) and estimate the solution from the graph. (e) Use calculus to find the critical number of the function in part (c). Then find the two numbers.

Short Answer

Expert verified
The two positive numbers whose sum is 110 and whose product is a maximum are both 55.

Step by step solution

01

Fill Table Analytically

Fill up the table choosing any value of x and finding the corresponding values of (110-x) and the product P=x(110-x). Repeat this process for four more rows for a total of six rows.
02

Graphical Representation

Use a graphing utility to generate additional rows of the table, which could help in estimating the solution. It's not necessary to mention specific numbers here because the graphical representation may vary depending on the used tool.
03

Write Function of Product P

Express the product P as a function of \(x\). As both numbers sum up to 110, the second number is (110-x). So, \(P=x(110-x)\) is the function to work with.
04

Graph Function and Estimate Solution

Again, use the graphing utility to graph the function \(P=x(110-x)\) and estimate the solution from the graph. Observe the peak of the function graph, which represents its maximum point, thus maximizing the product.
05

Find Critical Number using Calculus

Derive the function P to get \(P'(x) = 110 - 2x\). Set the derivative equal to zero to find the critical number \(x\), that is \(110 - 2x = 0\). Solve it to get \(x = 55\). This gives the critical number where the function reaches its maximum.
06

Find the Two Numbers

The two numbers are \(x\) and \(110 - x\). Thus, substituting \(x = 55\), the two numbers are 55 and 55 (55 + 55 = 110)

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