Problem 94 - Intermediate Algebra

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 maximum number of deliveries is 2743 per day.

Step 1: Understand the Problem

We are given the cost function \(C(x) = 80 \sqrt[3]{x} + 500\) and need to find the maximum \(x\) such that \(C(x) < 1620\).

Step 2: Setup the Inequality

Write the inequality for the problem: \(80 \sqrt[3]{x} + 500 < 1620\).

Step 3: Isolate the Cube Root Term

Subtract 500 from both sides to get: \(80 \sqrt[3]{x} < 1120\).

Step 4: Solve for the Cube Root

Divide both sides by 80 to get: \(\sqrt[3]{x} < 14\).

Step 5: Remove the Cube Root

Cube both sides to solve for \(x\): \(x < 14^3\).

Step 6: Calculate the Cube

Compute \(14^3 = 2744\). Thus, \(x < 2744\).

Step 7: Determine the Maximum Integer

Since \(x\) must be an integer, the greatest number of deliveries is 2743.

Key Concepts

Inequalities

In solving equations, inequalities are vital. They help us understand relationships where values aren't exactly equal but are rather one-sided. In our problem, we were given an inequality:

• \(80 \sqrt[3]{x} + 500 < 1620\)

We're to find the maximum number of deliveries that make this inequality true. It shows that for some number of deliveries, the cost remains below a certain threshold, in this case, \$1620\.
The process involves simplifying the inequality step by step until the variable is isolated.
We simplified by subtracting 500 and dividing by 80, making it easier to see the maximum value for \(x\). Remember, solving inequalities is not just about finding solutions but ensuring those solutions make sense given initial conditions.

Cost Functions

Cost functions are mathematical expressions that represent how a cost changes with varying levels of an activity.
Our problem's cost function is \(C(x)=80 \sqrt[3]{x}+500\).
This signifies that the cost depends on the cube root of the number of deliveries made, multiplied by 80, and adds a constant, 500.

• The constant represents fixed costs, which do not change regardless of the number of deliveries.
• The term involving \(\sqrt[3]{x}\) accounts for variable costs, capturing how expenses increase as delivery numbers rise.

Interpreting cost functions helps understand economic dynamics, showing how different factors interact.
It reflects real-world scenarios, like managing delivery services, offering insights on balancing operations within financial constraints.

Integer Solutions

When problems request integer solutions, they seek whole numbers, as fractional values aren't suitable. In our problem, even though we computed \(x < 2744\), the actual solution needed is the greatest integer less than 2744, which is 2743.

• Integer solutions are common in real-life scenarios where partial values have no meaning.
• Whole numbers simplify some models, like counting deliveries or customers accurately.

Understanding the need for integer solutions ensures effective decision-making. It's crucial where discrete values hold practical relevance, aiding in achieving realistic outcomes in problem-solving.