Question

Suppose that the daily cost \(C\) of manufacturing bicycles is given by \(C(x)=80 x+5000 .\) Then the average daily cost \(\bar{C}\) is given by \(\bar{C}(x)=\frac{80 x+5000}{x} .\) How many bicycles must be produced each day for the average cost to be no more than \(\$ 100 ?\)

Short answer

At least 250 bicycles must be produced each day.

Step-by-step solution

Set Up the Inequality

The problem asks for the number of bicycles, denoted as x, that must be produced for the average cost to be no more than $100. Therefore, set up the inequality:\[ \bar{C}(x) \leq 100 \]

Substitute the Average Cost Formula

Given that \( \bar{C}(x) = \frac{80x + 5000}{x} \), substitute this into the inequality:\[ \frac{80x + 5000}{x} \leq 100 \]

Solve the Inequality

Multiply both sides by x to clear the fraction:\[ 80x + 5000 \leq 100x \]Subtract 80x from both sides:\[ 5000 \leq 20x \]Finally, divide both sides by 20:\[ x \geq 250 \]

Key concepts

inequality in algebra

Inequality in algebra helps us determine the range of values for which a condition holds true. In this exercise, we are working with a situation where we want a daily average cost to not exceed 100 dollars. The inequality we form is \[ \bar{C}(x) \leq 100 \]. This means we need to find the values of x (bicycles produced) that satisfy this inequality.

The steps to solve an inequality generally involve:

• Setting up the inequality.
• Substituting known values into the inequality.
• Simplifying the inequality by performing algebraic operations like addition, subtraction, multiplication, or division.

In this problem, we used these steps to find the minimum number of bicycles that need to be produced to ensure the average cost doesn't exceed a specific amount.

cost function

The cost function represents the total cost of producing a certain number of goods. It usually depends on both the variable cost (which changes with the quantity produced) and the fixed cost (which doesn't change with the quantity produced).

In this exercise, the cost function is given by \( C(x) = 80x + 5000 \). Here:

• \[ 80x \] is the variable cost, as it changes with the number of bicycles produced (x).
• \[ 5000 \] is the fixed cost, which remains constant regardless of the number of bicycles produced.

The average daily cost function \[ \bar{C}(x) \] is the total cost function divided by the number of bicycles produced, \[ \bar{C}(x) = \frac{80x + 5000}{x} \]. This helps in assessing the unit cost when production levels vary.

division and multiplication in algebra

In algebra, division and multiplication are fundamental operations. They often help us solve equations and inequalities. In the step-by-step solution of this problem, we use both:

1. **Multiplication**: To clear the fraction in the inequality \[\frac{80x + 5000}{x} \leq 100 \], we multiply both sides by x:

• \[ \left(\frac{80x + 5000}{x}\right) \cdot x \leq 100 \cdot x \]
• Simplifying, we get: \[ 80x + 5000 \leq 100x \]

2. **Subtraction and Division**: To isolate x, we first subtract 80x from both sides:

• \[ 5000 \leq 100x - 80x \]
• This reduces to: \[5000 \leq 20x \]

Finally, divide both sides by 20:

• \[ \frac{5000}{20} \leq x \]
• \[ x \geq 250 \]

These operations help simplify the problem, making it easier to find the minimum number of bicycles produced to meet the given criteria.