Problem 89 - Algebra A Combined Function

Question

The cost \(C(x)\) in dollars per day to operate a small delivery service is given by \(C(x)=80 \sqrt[3]{x}+500,\) where \(x\) is the number of deliveries per day. In July, the manager decides that it is necessary to keep delivery costs below \(\$ 1620.00 .\) Find the greatest number of deliveries this company can make per day and still keep overhead below \(\$ 1620.00\)

Step-by-Step Solution

Answer

The greatest number of deliveries is 2743 per day.

Step 1: Understand the Problem

We need to find the maximum number of deliveries the company can make per day such that the cost per day remains below $1620. We have the cost equation \(C(x) = 80 \sqrt[3]{x} + 500\).

Step 2: Set Up the Inequality

The problem requires finding \(x\) such that the cost \(C(x) < 1620\). Therefore, we set up the inequality: \[80 \sqrt[3]{x} + 500 < 1620.\]

Step 3: Isolate the Cube Root

First, subtract 500 from both sides of the inequality: \[80 \sqrt[3]{x} < 1620 - 500.\] Simplifying the right side gives: \[80 \sqrt[3]{x} < 1120.\]

Step 4: Solve for Cube Root of x

Divide both sides by 80 to isolate the cube root: \[\sqrt[3]{x} < \frac{1120}{80}.\] Simplifying gives: \[\sqrt[3]{x} < 14.\]

Step 5: Solve for x

To solve for \(x\), cube both sides of the inequality: \[x < 14^3.\] Calculating \(14^3\) gives \(2744\). Thus, \[x < 2744.\]

Step 6: Interpret the Solution

Since \(x\) must be an integer number of deliveries, the maximum number of deliveries \(x\) that can be made without exceeding the cost is 2743.

Key Concepts

Inequalities

Inequalities are mathematical expressions used to indicate that one quantity is larger or smaller than another. In this exercise, we use inequalities to express the condition that the delivery costs must remain below $1620. We write this inequality as \(80 \sqrt[3]{x} + 500 < 1620\).

This inequality tells us about the relationship between the number of deliveries, \(x\), and the cost. The goal is to find the maximum value of \(x\) for which the inequality holds true, ensuring costs don't exceed the set limit.

To solve this, we manipulate the inequality step by step:

• Start by subtracting 500 from both sides so that we deal directly with the component affected by the deliveries.
• Next, divide both sides by 80 to isolate the cube root term, \(\sqrt[3]{x}\).
• Finally, solve for \(x\) by cubing both sides of the inequality.

This process allows us to determine that the maximum integer value of \(x\) is 2743.

Cost Analysis

Cost analysis is a crucial aspect of operations as it involves understanding and controlling expenditures. In this scenario, cost analysis helps in assessing the operating costs of a delivery service. The manager wants to ensure that the daily costs do not exceed $1620, based on the function \(C(x) = 80 \sqrt[3]{x} + 500\).

The function shows that costs increase as the number of deliveries goes up. The main goal is to determine how many deliveries can be made per day within the specified budget:

• The fixed part, 500, represents costs that do not change with delivery quantity, like rent or salaries.
• The variable part, \(80 \sqrt[3]{x}\), changes based on the number of deliveries, accounting for expenses like fuel or wages, which directly relate to delivery numbers.

By analyzing these components, businesses can make informed decisions on optimizing operations without compromising financial goals.

Operations Research

Operations research is a field of study focused on the optimization of complex processes or systems. It uses mathematical models and analytical methods to improve decision-making and efficiency. In this exercise, the delivery company applies operations research principles to optimize delivery numbers while managing costs.

By constructing the cost function \(C(x) = 80 \sqrt[3]{x} + 500\), the company can assess how changing delivery numbers affects overall expenses. Using the method of inequalities, the manager effectively determines an optimal delivery number within the budget constraint of $1620.

This process illustrates core ideas in operations research:

• Utilizing mathematical models to represent real-world situations.
• Applying analytical techniques to find optimal solutions and improve operations.
• Ensuring resource utilization is efficient while keeping costs under control.

Such analyses allow companies to maintain service quality while adhering to budget constraints, ultimately contributing to better strategic planning and resource management.