The simplest cost function is the linear cost function, C(x)= mx + b where the y intercept & represents the fixed costs of operating a business and the slope m represents the cost of each item produced. Suppose that a calculator manufacturer has daily fixed costs of $1000 and each calculator costs $10 to manufacture
(a) Write a linear model that expresses the cost C of manufacturing x calculators in a day.
(b) Graph the model.
(c) What is the cost of manufacturing 250 calculators in a day?
(d) How many calculators could be manufactured for $4500
The correct answer and explanation is :
Let’s address each part of the problem step by step:
(a) Write a linear model that expresses the cost C of manufacturing x calculators in a day.
A linear cost function can be expressed as:
[ C(x) = mx + b ]
Where:
- ( m ) is the variable cost per unit (the cost to produce each calculator).
- ( b ) is the fixed cost (the cost that does not change with the number of calculators produced).
Given that each calculator costs $10 to manufacture, the variable cost per unit is $10, so ( m = 10 ). The daily fixed costs are $1000, so ( b = 1000 ). Therefore, the linear cost function is:
[ C(x) = 10x + 1000 ]
(b) Graph the model.
To graph the cost function ( C(x) = 10x + 1000 ), plot the following points:
- When ( x = 0 ):
- ( C(0) = 10(0) + 1000 = 1000 )
- Point: (0, 1000)
- When ( x = 100 ):
- ( C(100) = 10(100) + 1000 = 2000 )
- Point: (100, 2000)
- When ( x = 200 ):
- ( C(200) = 10(200) + 1000 = 3000 )
- Point: (200, 3000)
Plotting these points and drawing a straight line through them will give the graph of the cost function. The line will have a slope of 10 (indicating that each additional calculator increases the total cost by $10) and a y-intercept of 1000 (representing the fixed daily costs).
(c) What is the cost of manufacturing 250 calculators in a day?
Using the cost function ( C(x) = 10x + 1000 ):
[ C(250) = 10(250) + 1000 = 2500 + 1000 = 3500 ]
Therefore, the cost of manufacturing 250 calculators in a day is $3500.
(d) How many calculators could be manufactured for $4500?
Using the cost function ( C(x) = 10x + 1000 ):
Set ( C(x) = 4500 ):
[ 4500 = 10x + 1000 ]
Subtract 1000 from both sides:
[ 3500 = 10x ]
Divide both sides by 10:
[ x = 350 ]
Therefore, 350 calculators could be manufactured for $4500.
Explanation:
In this scenario, the total manufacturing cost consists of two components: fixed costs and variable costs.
- Fixed Costs: These are costs that do not change with the level of production. In this case, the fixed daily costs are $1000, which represent expenses that occur regardless of how many calculators are produced.
- Variable Costs: These are costs that vary directly with the number of units produced. Here, each calculator costs $10 to manufacture.
The total cost function combines these two components:
[ C(x) = 10x + 1000 ]
This function allows us to calculate the total cost for any given number of calculators produced.
- Part (c): To find the cost of manufacturing 250 calculators, we substitute ( x = 250 ) into the cost function: [ C(250) = 10(250) + 1000 = 2500 + 1000 = 3500 ] This calculation shows that producing 250 calculators would incur a total cost of $3500.
- Part (d): To determine how many calculators can be produced for $4500, we set the cost function equal to 4500 and solve for ( x ): [ 4500 = 10x + 1000 ] Subtracting 1000 from both sides gives: [ 3500 = 10x ] Dividing both sides by 10 results in: [ x = 350 ] This means that with a budget of $4500, the company can manufacture 350 calculators.
Understanding the structure of the cost function is crucial for businesses to make informed decisions about production levels and budgeting. By analyzing fixed and variable costs, companies can optimize their operations to achieve desired profit margins and cost efficiencies.