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Chapter 2: Problem 26

Let \(Q(x)\) denote the total cost of manufacturing \(x\) units of some product. Then \(C(x)\) is an increasing function for all \(x\). For small values of \(x\), the rate of increase of \(C(x)\) decreases (because of the savings that are possible with "mass production"). Eventually, however, for large values of \(x\), the cost \(Q(x)\) increases at an increasing rate. (This happens when production facilities are strained and become less efficient.) Sketch a graph that could represent \(Q(x).\)

Short Answer

Expert verified
The graph of \( Q(x) \) initially steepens, then flattens, and finally steepens again.

Step by step solution

01

Understand the Given Information

The function \( Q(x) \) represents the total cost of manufacturing \( x \) units. It is stated that \( Q(x) \) is an increasing function, meaning that as \( x \) increases, \( Q(x) \) also increases.
02

Analyze the Behavior for Small Values of \( x \)

For small values of \( x \), the rate at which \( Q(x) \) increases decreases. This implies that the graph of \( Q(x) \) starts off steep and then starts to flatten out due to savings from mass production.
03

Analyze the Behavior for Large Values of \( x \)

For large values of \( x \), \( Q(x) \) starts to increase at an increasing rate. This would mean that after a certain point, the graph of \( Q(x) \) starts to steepen again due to inefficiencies in production.
04

Sketch the Graph

Start by drawing a plot with \( x \) on the horizontal axis and \( Q(x) \) on the vertical axis. Initially, draw a curve that rises steeply but then gradually flattens. After reaching a point, make the curve steepen again, representing the increasing rate of cost for larger values of \( x \).

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Key Concepts

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

increasing function
Understanding that the cost function, denoted as \( Q(x) \), is an increasing function is fundamental in manufacturing. This means that as the production of units \( x \) increases, the total cost \( Q(x) \) also increases. Visually, if you were to graph this function, you'd observe that it trends upwards from left to right. This steady upwards movement indicates that producing more units always results in higher costs鈥攁 concept easy to grasp but important to remember. Whether it's due to raw materials, labor, or additional logistical needs, every new unit adds to the total cost.
production cost
The term 'production cost' encompasses all expenses incurred in manufacturing goods. It includes:
  • Raw materials
  • Labor costs
  • Machine operation and maintenance
  • Utilities like electricity
  • Other operational costs
At small production levels, costs per unit can be relatively high because fixed costs are spread over fewer units. However, as production ramps up, these fixed costs are distributed over more units, leading to a decrease in the average cost per unit. This is why the cost function \( Q(x) \) initially increases at a decreasing rate.
economies of scale
Economies of scale refer to the cost advantages that enterprises obtain due to their scale of operation. As the production volume increases, the average cost per unit typically decreases. This concept explains why, for small values of \( x \), the rate at which \( Q(x) \) increases slows down. Mass production allows for:
  • Bulk purchasing discounts
  • Efficient use of machinery and infrastructure
  • Lower costs in per-unit production
These factors contribute to the graph of \( Q(x) \) flattening out as \( x \) increases initially.
inefficiencies in production
While economies of scale benefit lower production costs initially, inefficiencies in production eventually kick in, especially at higher levels of output. These inefficiencies can arise from:
  • Overworked machinery and labor force
  • Increased likelihood of equipment failures
  • Logistical complications in managing larger outputs
Such inefficiencies cause the cost function \( Q(x) \) to steepen as \( x \) becomes large, indicating that the rate at which costs increase starts to grow. Visually, this means that the graph of \( Q(x) \) will begin to rise more sharply after a certain point.

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Most popular questions from this chapter

Let \(s(t)\) be the distance (in \(\mathrm{feet}\) ) traveled by a parachutist after \(t\) seconds from the time of opening the chute, and suppose that \(s(t)\) has the line \(y=-15 t+10\) as an asymptote. What does this imply about the velocity of the parachutist? [Note: Distance traveled downward is given a negative value \(.]\)

A California distributor of sporting equipment expects to sell 10,000 cases of tennis balls during the coming year at a steady rate. Yearly carrying costs (to be computed on the average number of cases in stock during the year) are \(\$ 10\) per case, and the cost of placing an order with the manufacturer is \(\$ 80.\) (a) Find the inventory cost incurred if the distributor orders 500 cases at a time during the year. (b) Determine the economic order quantity, that is, the order quantity that minimizes the inventory cost.

A savings and loan association estimates that the amount of money on deposit will be 1 million times the percentage rate of interest. For instance, a \(4 \%\) interest rate will generate \(\$ 4\) million in deposits. If the savings and loan association can loan all the money it takes in at \(10 \%\) interest. what interest rate on deposits generates the greatest profit?

A store manager wants to establish an optimal inventory policy for an item. Sales are expected to be at a steady rate and should total \(Q\) items sold during the year. Each time an order is placed a cost of \(h\) dollars is incurred. Carrying costs for the year will be \(s\) dollars per item, to be figured on the average number of items in storage during the year. Show that the total inventory cost is minimized when each order calls for \(\sqrt{2 h Q / s}\) items.

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