Question

Manufacturing cost The weekly production cost \(C\) of manufacturing \(x\) watches is given by the formula \(C=4000+2 x,\) where the variable \(C\) is in dollars. (a) What is the cost of producing 1000 watches? (b) What is the cost of producing 2000 watches?

Short answer

The cost to produce 1000 watches is \$6000, and the cost to produce 2000 watches is \$8000.

Step-by-step solution

Identify the given formula

The cost formula given is:\[ C = 4000 + 2x \]where \( C \) is the production cost in dollars and \( x \) is the number of watches produced.

Substitute the value for 1000 watches

To find the cost of producing 1000 watches, substitute \( x = 1000 \) into the formula:\[ C = 4000 + 2(1000) \]

Calculate the cost for 1000 watches

Simplify the equation:\[ C = 4000 + 2000 = 6000 \]So, the cost of producing 1000 watches is \$6000.

Substitute the value for 2000 watches

To find the cost of producing 2000 watches, substitute \( x = 2000 \) into the formula:\[ C = 4000 + 2(2000) \]

Calculate the cost for 2000 watches

Simplify the equation:\[ C = 4000 + 4000 = 8000 \]So, the cost of producing 2000 watches is \$8000.

Key concepts

linear equation

A linear equation is an equation that makes a straight line when graphed. In a real-world context, it often represents relationships where one quantity depends on another. For our manufacturing cost problem, the linear equation given is \[ C = 4000 + 2x \]. Here, \(C\) represents the total production cost and \(x\) stands for the number of watches produced. The equation shows a direct relationship between production cost and the number of units produced. The term \(4000\) is the fixed cost, which doesn't change regardless of how many watches are produced. The term \(2x\) indicates the variable cost, which changes depending on the number of watches produced.

substitution

Substitution is a fundamental technique in algebra, where you replace a variable with a given value to solve an equation. In our exercise, we need to find the production costs for specific numbers of watches. To find the cost for 1000 watches, we substitute \( x = 1000 \) in the equation \( C = 4000 + 2x \). Doing the math, we get: \[ \begin{align*} C &= 4000 + 2(1000) \ &= 4000 + 2000 \ &= 6000 \]. This shows that the cost of producing 1000 watches is \(6000. Similarly, for 2000 watches, substitute \( x = 2000 \): \[ \begin{align*} C &= 4000 + 2(2000) \ &= 4000 + 4000 \ &= 8000 \]. The cost of producing 2000 watches is \)8000.

cost analysis

Cost analysis involves evaluating all the costs involved in manufacturing to understand how they change with the production volume. In our case, the cost analysis can be performed using the given linear equation \( C = 4000 + 2x \). We need to separate the fixed costs and variable costs. The fixed cost here is \(4000, which remains constant irrespective of the number of watches produced. The variable cost is determined by the term \(2x \), meaning it costs \)2 to produce one watch. Understanding these components helps in pricing, budgeting, and financial planning. If we know the total production cost and the number of watches, we can also back-calculate to find the individual contribution of fixed and variable costs.

production cost

Production cost is the total expense incurred in manufacturing a product. For the watches in our example, the total cost is given by \( C = 4000 + 2x \), where \( 4000 \) is the fixed cost, and \( 2x \) represents variable costs depending on the number of watches produced. Fixed costs (\(4000) include expenses that do not change with production volume like rent, salaries, machinery, etc. Variable costs (\(2x\)) increase with the number of units produced. To determine the total production cost for different production levels, you simply substitute the value of \(x\) into the equation. For instance, for 1000 watches, the production cost is \)6000, and for 2000 watches, it is $8000. This linear model simplifies predicting future costs as production scales.