Question

Suppose that the government imposes a $$\$ 1000$$ -per-day tax on the bicycle manufacturer so that the daily cost \(C\) of manufacturing \(x\) bicycles is now given by \(C(x)=80 x+6000 .\) Now the average daily cost \(\bar{C}\) is given by \(\bar{C}(x)=\frac{80 x+6000}{x} .\) How many bicycles must be produced each day for the average cost to be no more than $$\$ 100 ?$$

Short answer

300 bicycles.

Step-by-step solution

Write down the formula

The average daily cost \(\bar{C}(x)\) is given by \(\bar{C}(x) = \frac{80x + 6000}{x}\).

Set the average cost equal to 100

We need to find the value of \x\ such that the average cost is no more than \$100\. Therefore, we set \(\bar{C}(x) = 100\): \[ 100 = \frac{80x + 6000}{x} \]

Solve for \x\

To solve for \x\, multiply both sides by \x\: \[ 100x = 80x + 6000 \]

Isolate \x\

Subtract \80x\ from both sides: \[ 100x - 80x = 6000 \] \[ 20x = 6000 \]

Divide by 20

Now solve for \x\ by dividing both sides by 20: \[ x = \frac{6000}{20} \] \[ x = 300 \]

Key concepts

cost function

A cost function, often denoted as C(x), helps in understanding the total cost incurred in manufacturing a certain number of goods, denoted as x. For this bicycle manufacturer, the daily cost is given by the formula: \[ C(x) = 80x + 6000 \] This formula includes two parts:

• Variable Cost: This is the cost per unit produced. In this case, it’s \$80\ per bicycle, so for x bicycles, it’s \$80x\.
• Fixed Cost: This is a constant cost that does not change with the number of units produced. Here, it’s \$6000\, which might include expenses like rent, salaries, and in this problem, a tax of \$1000\ per day.

Understanding the cost function is essential as it reflects how costs scale with production and helps in making key business decisions.

solving equations

Solving equations is a fundamental skill in algebra. In this exercise, we need to solve for \(x\), the number of bicycles. Let's break down the steps to solve our equation: First, consider the equation we derive from setting the average cost to 100:\[ 100 = \frac{80x + 6000}{x} \]To eliminate the fraction, multiply both sides by \(x\):\[100x = 80x + 6000\]Next, isolate \(x\) by subtracting \(80x\) from both sides:\[100x - 80x = 6000\]\[20x = 6000\]Finally, divide both sides by 20:\[x = \frac{6000}{20} = 300\]Your answer, \(x = 300\), means 300 bicycles must be produced for the average cost to be no more than $100.

average cost

Average cost, denoted as \(\bar{C}(x)\), is the cost per unit when considering both variable and fixed costs. It's crucial because it reflects how production scales with both types of costs. To find it, you divide the total cost \(C(x)\) by the number of units produced \(x\): \[\bar{C}(x) = \frac{C(x)}{x} = \frac{80x + 6000}{x}\]Simplifying, we get: \[\bar{C}(x) = 80 + \frac{6000}{x}\]Here:

• \(80\) represents the variable cost per unit.
• \(\frac{6000}{x}\) represents how the fixed cost spreads over the number of units produced.

As more bicycles are produced, the fixed cost per unit decreases, lowering the average cost.

tax impact on cost

Taxes are part of fixed costs that can significantly impact overall and average costs. In this exercise, a \$1000\ daily tax is added to the fixed costs. The modified cost function is: \[C(x) = 80x + 6000\]Where \$1000\ of the \$6000\ is due to the tax.With this tax, businesses must produce more to keep the average cost low. In our example, the producer needs to make at least 300 bicycles daily to keep the average cost at \$100\ or less. Without the tax, the cost function would be: \[C(x) = 80x + 5000\]Therefore, to keep average costs low, understanding all components of the cost, including taxes, is vital for effective budgeting and pricing strategies.