/*! 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 70 A dietitian designs a special di... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 70

A dietitian designs a special dietary supplement using two different foods. Each ounce of food X contains 20 units of calcium, 15 units of iron, and 10 units of vitamin B. Each ounce of food Y contains 10 units of calcium, 10 units of iron, and 20 units of vitamin B. The minimum daily requirements of the diet are 300 units of calcium, 150 units of iron, and 200 units of vitamin B. (a) Write a system of inequalities describing the different amounts of food \(X\) and food Y that can be used. (b) Sketch a graph of the region corresponding to the system in part (a). (c) Find two solutions of the system and interpret their meanings in the context of the problem.

Short Answer

Expert verified
The system of inequalities that describes the different amounts of food X and Y that can be used are: \(20x + 10y >= 300\), \(15x + 10y >= 150\), and \(10x + 20y >= 200\). The solutions to the system which corresponds to the amounts of foods X and Y (in ounces) that meet the daily nutritional requirements can be determined from the graphical representation of the inequalities.

Step by step solution

01

Identifying variables and setting up equations

Let's denote the quantities of food X and Y to be consumed as \(x\) and \(y\), respectively. Based on the given information, the following inequalities can be formed: \nFor calcium: \(20x + 10y >= 300 \)\nFor iron: \(15x + 10y >= 150 \)\nFor Vitamin B: \(10x + 20y >= 200 \)
02

Graphing the system of inequalities

To graph these inequalities, firstly, re-arrange each inequality in slope-intercept form (y = mx + c). Then plot each equation. All the regions that satisfy all the inequalities represent the possible combinations of amounts of foods X and Y that meet the daily nutritional requirements.
03

Finding and interpreting solutions

Pick any two points in the intersection region of the graph. Each point corresponds to a solution to the system, therefore each point represents a combination of foods X and Y that meets the requirements. For instance, a point (x, y) in the feasible region could be interpreted as consuming x ounces of food X and y ounces of food Y would meet the daily nutritional requirements.

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.

Nutritional Requirements
When creating a dietary plan, it's crucial to determine the nutritional needs of an individual. In this task, a dietitian needs to ensure that the chosen combination of foods meets the daily requirements of calcium, iron, and vitamin B. Understanding these requirements is important as deficiencies can lead to health issues. The problem outlines minimum daily needs:
  • 300 units of calcium
  • 150 units of iron
  • 200 units of vitamin B
The aim is to use a combination of food X and food Y to meet or exceed these minimum daily intakes. Each food item provides a different amount of nutrients per ounce:
  • Food X: 20 units of calcium, 15 units of iron, 10 units of vitamin B
  • Food Y: 10 units of calcium, 10 units of iron, 20 units of vitamin B
Understanding and analyzing these nutritional values helps us formulate a system of inequalities, which models the combinations of food servings that satisfy all these nutritional needs.
Graphing Inequalities
Graphing the system of inequalities provides a visual representation of possible solutions. To begin graphing, convert each inequality into a linear equation:
  • Calcium: \(20x + 10y \geq 300\)
  • Iron: \(15x + 10y \geq 150\)
  • Vitamin B: \(10x + 20y \geq 200\)
Rearrange these into the slope-intercept form which is \(y = mx + c\). Plot each line on a coordinate grid. The solution region, or feasible region, is where all conditions overlap. This region represents combinations of food X and food Y that meet all the nutritional requirements.
To highlight these regions:
  • After drawing each line, shade the region that satisfies the inequality.
  • The intersection where all shaded regions overlap is the feasible region.
Graphing aids in visualizing solutions that respect multiple dietary constraints simultaneously.
Solution Interpretation
Once the graph is complete, you can identify points within the feasible region. Each point is a potential solution, indicating a viable combination of food X and Y servings. For instance, choosing a point (3, 12) might indicate using 3 ounces of food X and 12 ounces of food Y. It's important to verify that these satisfy all inequalities:
  • Check that each nutritional component meets or exceeds its respective requirement when combined.
  • If the point lies in the feasible region, it confirms the nutritional needs are met.
Interpreting these solutions within the context of the problem ensures the dietary plan meets health recommendations. Remember, in real contexts, it's best to consider cost, availability, and more when interpreting these mathematical solutions into practical dietary plans.

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

Solve the system of linear equations and check any solutions algebraically. $$\left\\{\begin{array}{c} x\quad\quad+4 z=1 \\ x+y+10 z=10 \\ 2 x-y+2 z=-5 \end{array}\right.$$

A mixture of 5 pounds of fertilizer \(A\), 13 pounds of fertilizer \(\mathrm{B},\) and 4 pounds of fertilizer \(\mathrm{C}\) provides the optimal nutrients for a plant. Commercial brand X contains equal parts of fertilizer \(\mathrm{B}\) and fertilizer \(\mathrm{C}\). Commercial brand Y contains one part of fertilizer \(\mathrm{A}\) and two parts of fertilizer B. Commercial brand Z contains two parts of fertilizer \(\mathrm{A}\), five parts of fertilizer \(\mathrm{B},\) and two parts of fertilizer C. How much of each fertilizer brand is needed to obtain the desired mixture?

A person plans to invest up to \(\$ 20,000\) in two different interest-bearing accounts. Each account is to contain at least \(\$ 5000 .\) Moreover, the amount in one account should be at least twice the amount in the other account. Find and graph a system of inequalities to describe the various amounts that can be deposited in each account.

The linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the minimum and maximum values of the objective function (if possible) and where they occur. Objective function: \(z=x+y\) Constraints: $$\begin{aligned}x & \geq 0 \\\y & \geq 0 \\\\-x+y & \leq 1 \\\\-x+2 y & \leq 4\end{aligned}$$

Use a graphing utility to graph the region determined by the constraints. Then find the minimum and maximum values of the objective function and where they occur, subject to the constraints. Objective function: \(z=3 x+y\) Constraints: $$\begin{aligned}x & \geq 0 \\\y & \geq 0 \\ x+4 y & \leq 60 \\\3 x+2 y & \geq 48\end{aligned}$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

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.