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Chapter 1: Problem 11

Total cost: The total cost \(C\) for a manufacturer during a given time period is a function of the number \(N\) of items produced during that period. To determine a formula for the total cost, we need to know the manufacturer's fixed costs (covering things such as plant maintenance and insurance), as well as the cost for each unit produced, which is called the variable cost. To find the total cost, we multiply the variable cost by the number of items produced during that period and then add the fixed costs. Suppose that a manufacturer of widgets has fixed costs of \(\$ 9000\) per month and that the variable cost is \(\$ 15\) per widget (so it costs \(\$ 15\) to produce 1 widget). a. Use a formula to express the total cost \(C\) of this manufacturer in a month as a function of the number of widgets produced in a month. Be sure to state the units you use. b. Express using functional notation the total cost if there are 250 widgets produced in a month, and then calculate that value.

Short Answer

Expert verified
The total cost function is \( C(N) = 15N + 9000 \). For 250 widgets, the total cost is \( \$12750 \).

Step by step solution

01

Define the variables and constants

Identify the fixed costs and variable costs from the problem statement.- Fixed costs: \( \\(9000 \) per month - Variable cost per widget: \( \\)15 \) - Let \( N \) be the number of widgets produced in a month.
02

Formulate the total cost function

To express the total cost as a function of the number of widgets, use the formula:\[ C(N) = \text{Variable Cost} \times N + \text{Fixed Costs} \]Given:- Variable Cost per widget = \( \\(15 \)- Fixed Costs = \( \\)9000 \)Thus, the function becomes:\[ C(N) = 15N + 9000 \]
03

Calculate the total cost for 250 widgets

Substitute \( N = 250 \) into the total cost function:\[ C(250) = 15 \times 250 + 9000 \]
04

Simplify and find the result

Calculate the expression from Step 3:\[ C(250) = 3750 + 9000 = 12750 \]Thus, the total cost for producing 250 widgets is \( \$12750 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fixed Costs
Fixed costs are integral to understanding how the total cost for producing items is structured. These costs do not change based on the production level. For a manufacturing company, fixed costs refer to expenses that remain the same regardless of how many items are produced. Examples include rent for plant facilities, salaries of permanent staff, and insurance premiums.

In our exercise scenario, the fixed cost is given as $9,000 per month. This means that every month, regardless of whether the company produces 10 widgets or 10,000 widgets, the cost of $9,000 will be incurred.
  • These fixed costs are important to cover basic operational expenses even when production halts.
  • They ensure the company can maintain its operational capabilities and continue to produce when required.
Understanding fixed costs helps in planning and predicting long-term financial needs since these expenses are predictable and consistent each month.
Variable Costs
While fixed costs never change, variable costs fluctuate with the level of production. This means that the more widgets you produce, the higher the overall variable costs will be. Variable costs include expenses such as raw materials, direct labor, and other expenses that increase with production volume.

In the provided exercise, the variable cost is specified as $15 per widget. This indicates that for each additional widget produced, the company incurs an extra $15 in costs. Variable costs, when added up, help determine the total production cost depending on the output level.
  • They offer flexibility in the production process; if fewer items are needed, the costs decrease accordingly.
  • They help in allocating resources efficiently, focusing on variable factors to control total costs effectively.
Addressing variable costs allows manufacturers to optimize production and pricing strategies.
Functional Notation
Functional notation helps us express the relationship between different variables in a concise and understandable way. It is a mathematical shorthand that shows how one quantity depends on another. For costs, it shows how the total cost changes as the number of units produced changes.

In the exercise, the total cost is expressed as a function of the number of widgets produced: \[ C(N) = 15N + 9000 \] Here, \(C(N)\) represents the total cost as a function of \(N\), the number of widgets produced. If you want to find the cost of producing a specific number of widgets, you simply substitute \(N\) with that number.

For example, to find the cost for 250 widgets, substitute 250 into the function: \[ C(250) = 15 imes 250 + 9000 = 12750 \] The functional notation aids in performing calculations easily, ensuring clarity in expressing how total costs are impacted by changes in production.

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Most popular questions from this chapter

Preparing a letter, continued: This is a continuation of Exercise 6. You pay your secretary \(\$ 9.25\) per hour. A stamped envelope costs 50 cents, and regular stationery costs 3 cents per page, but fancy letterhead stationery costs 16 cents per page. Assume that a letter requires fancy letterhead stationery for the first page but that regular paper will suffice for the rest of the letter. a. How much does the stationery alone cost for a 3-page letter? b. How much does it cost to prepare and mail a 3 -page letter if your secretary spends 2 hours on typing and corrections? c. Use a formula to express the cost of the stationery alone for a letter as a function of the number of pages in the letter. Identify the function and each of the variables you use, and state the units. d. Use a formula to express the cost of preparing and mailing a letter as a function of thenumber of pages in the letter and the time it takes your secretary to type it. Identify the function and each of the variables you use, and state the units. e. Use the function you made in part \(d\) to find the cost of preparing and mailing a 2 -page letter that it takes your secretary 25 minutes to type.

Head and aquifers: This is a continuation of Exercise 21. In underground water supplies such as aquifers, the water normally permeates some other medium such as sand or gravel. The head for such water is determined by first drilling a well down to the water source. When the well reaches the aquifer, pressure causes the water to rise in the well. The head is the height to which the water rises. In this setting, we get the pressure using Pressure \(=\) Density \(\times 9.8 \times\) Head \(.\) Here density is in kilograms per cubic meter, head is in meters, and pressure is in newtons per square meter. (One newton is about a quarter of a pound.) A sandy layer of soil has been contaminated with a dangerous fluid at a density of 1050 kilograms per cubic meter. Below the sand there is a rock layer that contains water at a density of 990 kilograms per cubic meter. This aquifer feeds a city water supply. Test wells show that the head in the sand is \(4.3\) meters, whereas the head in the rock is \(4.4\) meters. A liquid will flow from higher pressure to lower pressure. Is there a danger that the city water supply will be polluted by the material in the sand layer?

Research project: For this project you are to find and describe a function that is commonly used. Find a patient person whose job is interesting to you. Ask that person what types of calculations he or she makes. These calculations could range from how many bricks to order for building a wall to lifetime wages lost for a wrongful-injury settlement to how much insulin to inject. Be creative and persistentdon't settle for "I look it up in a table." Write a description, in words, of the function and how it is calculated. Then write a formula for the function, carefully identifying variables and units.

Tubeworm: An article in Nature reports on a study of the growth rate and life span of a marine tubeworm. \({ }^{12}\) These tubeworms live near hydrocarbon seeps on the ocean floor and grow very slowly indeed. Collecting data for creatures at a depth of 550 meters is extremely difficult. But for tubeworms living on the Louisiana continental slope, scientists developed a model for the time \(T\) (measured in years) required for a tubeworm to reach a length of \(L\) meters. From this model the scientists concluded that this tubeworm is the longest-lived noncolonial marine invertebrate known. The model is $$ T=14 e^{1.4 L}-20 \text {. } $$ A tubeworm can grow to a length of 2 meters. How old is such a creature? (Round your answer to the nearest year.)

A population of deer: When a breeding group of animals is introduced into a restricted area such as a wildlife reserve, the population can be expected to grow rapidly at first but to level out when the population grows to near the maximum that the environment can support. Such growth is known as logistic population growth, and ecologists sometimes use a formula to describe it. The number \(N\) of deer present at time \(t\) (measured in years since the herd was introduced) on a certain wildlife reserve has been determined by ecologists to be given by the function $$ N=\frac{12.36}{0.03+0.55^{t}} $$ a. How many deer were initially on the reserve? b. Calculate \(N(10)\) and explain the meaning of the number you have calculated. c. Express the number of deer present after 15 years using functional notation, and then calculate it. d. How much increase in the deer population do you expect from the 10 th to the 15 th year?
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