/*! 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 48 Investments An investor has up t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 48

Investments An investor has up to $$\$ 450,000$$ to invest in two types of investments. Type A investments pay \(8 \%\) annually and type \(B\) pay \(14 \%\) annually. To have a well-balanced portfolio, the investor imposes the following conditions. At least one-half of the total portfolio is to be allocated to type A investments and at least one-fourth is to be allocated to type \(\mathrm{B}\) investments. What is the optimal amount that should be invested in each type of investment? What is the optimal return?

Short Answer

Expert verified
The optimal amount to be invested in each type of investment is \$300,000 in type A and \$150,000 in type B. The optimal return is \$45,000.

Step by step solution

01

Define the Variables

Let \( x \) be the amount invested in type A investment, \( y \) be the amount invested in type B investment.
02

Formulate the Linear Programming Problem

The objective function to maximize is the total return: \( R = 0.08x + 0.14y \) (given by the interest rates of the investments). The constraints are explicitly given in the problem: \( x \geq 0.5(x+y) \) (at least half of the money is invested in A), \( y \geq 0.25(x+y) \) (at least one-fourth is invested in B), and \( x+y \leq 450,000\) (the total investment amount is limited to $450,000).
03

Solve the Linear Programming Problem

By solving the equations: \( x \geq 0.5(x+y) \), \( y \geq 0.25(x+y) \), and \( x+y = 450,000 \), we can find the optimal \( x \) and \( y \) which maximize \( R \). The corner points of feasibility region are computed by setting the constraints to equality. We will find that the optimal solution lies at \( x = 300,000 \) and \( y = 150,000 \) by testing all corner points.
04

Calculate the Optimal Return

With the optimal values of \( x \) and \( y \), we can substitute these values into objective function to get the optimal return: \( R = 0.08(300,000) + 0.14(150,000) = 24,000 + 21,000 = \$45,000 \).

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.

Investment Optimization
Investment optimization is a critical process in the financial world, where investors are looking to allocate their resources in the most efficient and profitable way. In the context of linear programming, this involves selecting the best possible combination of investment options to maximize returns or minimize risks, subject to a set of constraints.

In the given exercise, an investor needs to distribute their funds between two types of investments, A and B, with different annual interest rates. Linear programming aids in determining the optimal investment in each type to achieve the highest possible annual return. By setting an objective function and applying constraints reflective of the investor's conditions, the linear program forms a blueprint that maps out the most beneficial investment strategy.

Understanding investment optimization within linear programming is imperative for students not only in academic exercises but also as a fundamental skill for future financial planning and management.
Objective Function
The Objective Function in linear programming is essentially the heart of the optimization problem. It is the formula that must be maximized or minimized given a set of constraints. In investment scenarios, the objective function usually represents the total return or profit that needs to be optimized.

In our example, the total annual return from investments, denoted as R, serves as the objective function. It is defined as a function of the amounts invested in types A and B, mathematically expressed as R = 0.08x + 0.14y. As the investor's goal is to maximize this return, the solution to the linear program will determine the values of x (amount invested in type A) and y (amount invested in type B) that lead to the highest possible R while adhering to the constraints imposed.

This objective function provides a clear target for the linear programming model and guides the decision-making process. Students should pay particular attention to correctly formulating this function, as it can dictate the success or failure of the entire optimization effort.
Constraints in Linear Programming
Constraints in linear programming are the conditions or limitations that must be considered when finding the optimal solution to an optimization problem. These can include budget limits, resource capacities, or specific requirements that need to be fulfilled.

In the exercise context, the constraints reflect the investor's strategy for a well-balanced portfolio: a minimum amount to be invested in each type of investment. They are mathematically formulated as inequalities which the variables must satisfy. Specifically, for our scenario the constraints are:
  • At least half of the total investment should be in type A: x ≥ 0.5(x+y)
  • At least one-quarter of the total investment should be in type B: y ≥ 0.25(x+y)
  • The sum of investments in types A and B must not exceed $450,000: x+y ≤ 450,000

Students need to comprehend that these constraints shape the feasible region, or the set of all possible solutions that satisfy the constraints. The optimal solution must lie within this region. Mastery in identifying and applying constraints is crucial for effectively solving linear programming problems and achieving optimal results.

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

Maximize the objective function subject to the constraints \(x+4 y \leq 20, x+y \leq 8\) \(3 x+2 y \leq 21, x \geq 0\), and \(y \geq 0\) $$z=12 x+5 y$$

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}x<2 y-y^{2} \\ 0

MAKE A DECISION: STOPPING DISTANCE In testing of the new braking system of an automobile, the speed (in miles per hour) and the stopping distance (in feet) were recorded in the table below. $$ \begin{array}{|c|c|} \hline \text { Speed, } x & \text { Stopping distance, } y \\ \hline 30 & 54 \\ \hline 40 & 116 \\ \hline 50 & 203 \\ \hline 60 & 315 \\ \hline 70 & 452 \\ \hline \end{array} $$ (a) Find the least squares regression parabola \(y=a x^{2}+b x+c\) for the data by solving the following system. \(\left\\{\begin{array}{r}5 c+250 b+13,500 a=1140 \\ 250 c+13,500 b+775,000 a=66,950 \\ 13,500 c+775,000 b+46,590,000 a=4,090,500\end{array}\right.\) (b) Use the regression feature of a graphing utility to check your answer to part (a). (c) A car design specification requires the car to stop within 520 feet when traveling 75 miles per hour. Does the new braking system meet this specification?

MAKE A DECISION: DIET SUPPLEMENT A dietitian designs a special diet supplement using two different foods. Each ounce of food \(\mathrm{X}\) contains 20 units of calcium, 10 units of iron, and 15 units of vitamin \(\mathrm{B}\). Each ounce of food \(\mathrm{Y}\) contains 15 units of calcium, 20 units of iron, and 20 units of vitamin \(\mathrm{B}\). The minimum daily requirements for the diet are 400 units of calcium, 250 units of iron, and 220 units of vitamin B. (a) Find a system of inequalities describing the different amounts of food \(\mathrm{X}\) and food \(\mathrm{Y}\) that the dietitian can use in the diet. (b) Sketch the graph of the system. (c) A nutritionist normally gives a patient 18 ounces of food \(\mathrm{X}\) and \(3.5\) ounces of food \(\mathrm{Y}\) per day. Supplies of food \(\mathrm{X}\) are running low. What other combinations of foods \(\mathrm{X}\) and \(\mathrm{Y}\) can be given to the patient to meet the minimum daily requirements?

MAKE A DECISION: DIET SUPPLEMENT A dietitian designs a special diet supplement using two different foods. Each ounce of food \(\mathrm{X}\) contains 12 units of calcium, 10 units of iron, and 20 units of vitamin \(\mathrm{B}\). Each ounce of food \(\mathrm{Y}\) contains 15 units of calcium, 20 units of iron, and 12 units of vitamin B. The minimum daily requirements for the diet are 300 units of calcium, 280 units of iron, and 300 units of vitamin \(\mathrm{B}\). (a) Find a system of inequalities describing the different amounts of food \(\mathrm{X}\) and food \(\mathrm{Y}\) that the dietitian can use in the diet. (b) Sketch the graph of the system. (c) A nutritionist normally gives a patient 10 ounces of food \(\mathrm{X}\) and 12 ounces of food \(\mathrm{Y}\) per day. Supplies of food \(\mathrm{Y}\) are running low. What other combinations of foods \(\mathrm{X}\) and \(\mathrm{Y}\) can be given to the patient to meet the minimum daily requirements?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

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.