/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 We are to buy quantities of two ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 6

We are to buy quantities of two items: \(n_{1}\) units of item 1 and \(n_{2}\) units of item \(2 .\) a. If item 1 costs \(\$ 3.50\) per unit and item 2 costs \(\$ 2.80\) per unit, find a formula that gives the total \(\operatorname{cost} C\), in dollars, of the purchase. b. An isocost equation shows the relationship between the number of units of each of two items to be purchased when the total purchase price is predetermined. If the total purchase price is predetermined to be \(C=162.40\) dollars, find the isocost equation for items 1 and 2 from part a. c. Solve for \(n_{1}\) the isocost equation you found in part b. d. Using the equation from part c, determine how many units of item 1 are to be purchased if 18 units of item 2 are purchased.

Short Answer

Expert verified
a. \( C = 3.5n_1 + 2.8n_2 \); b. \( 3.5n_1 + 2.8n_2 = 162.40 \); c. \( n_1 = \frac{162.40 - 2.8n_2}{3.5} \); d. \( n_1 = 32 \) units for \( n_2 = 18 \).

Step by step solution

01

Setup Cost Formula

To find the total cost \(C\), we use the formula incorporating the costs of both items. Multiply the cost per unit of item 1 by the number of units \(n_1\), and do the same for item 2. The formula is: \[ C = 3.5n_1 + 2.8n_2 \]
02

Establish Isocost Equation

With the predetermined total cost \( C = 162.40 \), set the formula from Step 1 equal to this value. The isocost equation is:\[ 3.5n_1 + 2.8n_2 = 162.40 \]
03

Solve the Isocost Equation for \(n_1\)

To express \( n_1 \) in terms of \( n_2 \), we need to rearrange the isocost equation from Step 2. Subtract \( 2.8n_2 \) from both sides and then divide by 3.5:\[ n_1 = \frac{162.40 - 2.8n_2}{3.5} \]
04

Calculate \(n_1\) for Specific \(n_2\)

Using the equation from Step 3, substitute \( n_2 = 18 \) into the expression for \( n_1 \):\[ n_1 = \frac{162.40 - 2.8 \times 18}{3.5} \] Calculate the expression:\[ n_1 = \frac{162.40 - 50.4}{3.5} = \frac{112}{3.5} \approx 32 \] So, \( n_1 \approx 32 \) units.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Cost Analysis
When you go shopping, especially for multiple items, it's crucial to understand how each item contributes to the total cost. This is what cost analysis is all about. Cost analysis involves breaking down the total cost of a purchase into the individual costs of each item. This makes it easier to understand where your money is going and allows you to make informed purchasing decisions.

In our original problem, we were tasked with analyzing the cost of buying two items. Each item has its own price per unit: \(3.50 for item 1 and \)2.80 for item 2. Cost analysis helps us see how buying more or less of each item affects our total spending.

The formula for the total cost, as derived in the exercise, is:
  • \( C = 3.5n_1 + 2.8n_2 \)
Here, \( n_1 \) and \( n_2 \) represent the number of units purchased for item 1 and item 2, respectively. By plugging different values of \( n_1 \) and \( n_2 \) into this formula, we can see how changing the quantity of each item changes the total cost. Cost analysis is an essential skill in both personal finance and business settings, making it universally valuable.
Isocost Equation
The concept of an isocost equation is fundamental in economic analysis, particularly when analyzing constraints under which decisions need to be made. An isocost equation shows the relationship between the quantities of two or more goods that a consumer can buy while staying within a predetermined budget.

In the exercise, we derived an isocost equation based on a total cost (or budget) of \(162.40. The isocost equation derived from our cost formula is:
  • \( 3.5n_1 + 2.8n_2 = 162.40 \)
This equation represents all combinations of \( n_1 \) and \( n_2 \) that total \)162.40 when purchased at their respective unit prices.

Understanding this concept aids in making budget decisions. If your budget is fixed, this equation helps determine feasible combinations of items you can purchase, helping to allocate resources efficiently. Keep in mind that variations in unit prices would necessitate a new isocost line, maintaining flexibility in planning.
Problem Solving in Algebra
Algebra is a powerful toolbox for solving problems, especially those involving unknown values like in our exercise. The principles of algebra allow us to create equations based on known constants and variables, and then manipulate these equations to find the values of unknown quantities.

In the exercise, our task was to solve for \( n_1 \), representing the number of units of item 1, with given conditions. This was achieved by rearranging the isocost equation:
  • Start with the isocost equation: \( 3.5n_1 + 2.8n_2 = 162.40 \)
  • Rearrange to solve for \( n_1 \): \( n_1 = \frac{162.40 - 2.8n_2}{3.5} \)
Then, by substituting specific values for \( n_2 \), such as 18, we calculate \( n_1 \). This process exemplifies the power of algebra in isolating variables and solving real-world problems involving constraints and multiple variables.

By mastering these algebraic techniques, students are better equipped to tackle diverse problems, not just in math, but in everyday decision-making scenarios that involve logical reasoning and arithmetic calculations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

This is a continuation of Exercise 7. In many situations, the number of possibilities is not affected by order. For example, if a group of 4 people is selected from a group of 20 to go on a trip, then the order of selection does not matter. In general, the number \(C\) of ways to select a group of \(k\) things from a group of \(n\) things is given by $$ C=\frac{n !}{k !(n-k) !} $$ if \(k\) is not greater than \(n\). a. How many different groups of 4 people could be selected from a group of 20 to go on a trip? b. How many groups of 16 could be selected from a group of 20 ? c. Your answers in parts a and b should have been the same. Explain why this is true. d. What group size chosen from among 20 people will result in the largest number of possibilities? How many possibilities are there for this group size?

The background for this exercise can be found in Exercises 11, 12, 13, and 14 in Section 1.4. A manufacturer of widgets has fixed costs of \(\$ 1200\) per month, and the variable cost is \(\$ 40\) per widget (so it costs \(\$ 40\) to produce 1 widget). Let \(N\) be the number of widgets produced in a month. a. Find a formula for the manufacturer's total cost \(C\) as a function of \(N\). b. The highest price \(p\), in dollars, of a widget at which \(N\) widgets can be sold is given by the formula \(p=53-0.01 N\). Using this, find a formula for the total revenue \(R\) as a function of \(N\). c. Use your answers to parts a and \(b\) to find \(a\) formula for the profit \(P\) of this manufacturer as a function of \(N\). d. Use your formula from part c to determine the two break-even points for this manufacturer. Assume here that the manufacturer produces the widgets in blocks of 50 , so a table setup showing \(N\) in multiples of 50 is appropriate. e. Use your formula from part c to determine the production level at which profit is maximized if the manufacturer can produce at most 1500 widgets in a month. As in part d, assume that the manufacturer produces the widgets in blocks of 50 .

Friction loss in fire hoses: When water flows inside a hose, the contact of the water with the wall of the hose causes a drop in pressure from the pumper to the nozzle. This drop is known as friction loss. Although it has come under criticism for lack of accuracy, the most commonly used method for calculating friction loss for flows under 100 gallons per minute uses what is called the underwriter's formula: $$ F=\left(2\left(\frac{Q}{100}\right)^{2}+\frac{Q}{200}\right)\left(\frac{L}{100}\right)\left(\frac{2.5}{D}\right)^{5} $$ Here \(F\) is the friction loss in pounds per square inch, \(Q\) is the flow rate in gallons per minute, \(L\) is the length of the hose in feet, and \(D\) is the diameter of the hose in inches. a. In a 500 -foot hose of diameter \(1.5\) inches, the friction loss is 96 pounds per square inch. What is the flow rate? b. In a 500 -foot hose, the friction loss is 80 pounds per square inch when water flows at 65 gallons per minute. What is the diameter of the hose? Round your answer to the nearest \(\frac{1}{8}\) inch.

You have 16 miles of fence that you will use to enclose a rectangular field. a. Draw a picture to show that you can arrange the 16 miles of fence into a rectangle of width 3 miles and length 5 miles. What is the area of this rectangle? b. Draw a picture to show that you can arrange the 16 miles of fence into a rectangle of width 2 miles and length 6 miles. What is the area of this rectangle?

In 1933, Riatt found that the length \(L\) of haddock in centimeters as a function of the age \(t\) in years is given approximately by the formula $$ L=53-42.82 \times 0.82^{t} . $$ a. Calculate \(L(4)\) and explain what it means. b. Compare the average yearly rate of growth in length from age 5 to 10 years with the average yearly rate of growth from age 15 to 20 years. Explain in practical terms what this tells you about the way haddock grow. c. What is the longest haddock you would expect to find anywhere?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.