/*! 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 57 Solve each problem. Plate-Glas... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 57

Solve each problem. Plate-Glass Sales The amount of plate-glass sales \(S\) (in millions of dollars) can be affected by the number of new building contracts \(B\) issued (in millions) and automobiles \(A\) produced (in millions). A plate-glass company in California wants to forecast future sales by using the past three years of sales. The totals for the three years are given in the table. To describe the relationship among these variables, we can use the equation $$ S=a+b A+c B $$. where the coefficients \(a, b,\) and \(c\) are constants that must be determined before the equation can be used. (Source: Makridakis, S., and S. Wheelwright, Forecasting Methods for Management, John Wiley and Sons.) (a) Substitute the values for \(S, A,\) and \(B\) for each year from the table into the equation \(S=a+b A+c B,\) and obtain three linear equations involving \(a, b,\) and \(c\) (b) Use a graphing calculator to solve this linear system for \(a, b,\) and \(c .\) Use matrix inverse methods. (c) Write the equation for \(S\) using these values for the coefficients. $$\begin{array}{|c|c|c|} \hline S & A & B \\ \hline 602.7 & 5.543 & 37.14 \\\ \hline 656.7 & 6.933 & 41.30 \\ \hline 778.5 & 7.638 & 45.62 \\ \hline \end{array}$$ (d) For the next year it is estimated that \(A=7.752\) and \(B=47.38 .\) Predict \(S .\) (The actual value for \(S\) was \(877.6 .\) ) (e) It is predicted that in 6 yr, \(A=8.9\) and \(B=66.25 .\) Find the value of \(S\) in this situation and discuss its validity.

Short Answer

Expert verified
Formulate and solve the system of equations, write the found relationship, and use it to predict future sales.

Step by step solution

01

- Formulate Equations

Substitute the given values for each year into the equation \( S=a + bA + cB \): Year 1: \( 602.7 = a + 5.543b + 37.14c \) Year 2: \( 656.7 = a + 6.933b + 41.30c \) Year 3: \( 778.5 = a + 7.638b + 45.62c \)
02

- Matrix Formulation

Convert these equations into a matrix form \[ \begin{bmatrix} 1 & 5.543 & 37.14 \ 1 & 6.933 & 41.30 \ 1 & 7.638 & 45.62 \ \textend{bmatrix} \begin{bmatrix} a \ b \ c \ \textend{bmatrix} = \begin{bmatrix} 602.7 \ 656.7 \ 778.5 \ \textend{bmatrix} \]
03

- Solve for a, b, and c using Matrix Inverse Methods

Using a graphing calculator, solve the matrix equation above to find values of \(a\), \(b\), and \(c\). The matrix equation is of the form\[ AX = B \] where \[ A = \begin{bmatrix} 1 & 5.543 & 37.14 \ 1 & 6.933 & 41.30 \ 1 & 7.638 & 45.62 \ \textend{bmatrix}, X = \begin{bmatrix} a \ b \ c \ \textend{bmatrix}, B = \begin{bmatrix} 602.7 \ 656.7 \ 778.5 \ \textend{bmatrix}. \] Solve for \(X\) by computing \[ X = A^{-1}B \].
04

- Write the Equation for S

Substitute the found values of \(a\), \(b\), and \(c\) into the equation \( S = a + bA + cB \). For example, if the calculated values are \( a=10 \), \( b=20 \), and \( c=30 \), the equation becomes \( S = 10 + 20A + 30B \).
05

- Predict S for Given Values of A and B Next Year

Substitute the estimated values for next year \( A = 7.752 \) and \( B = 47.38 \) into the equation \( S = a + bA + cB \). Calculate \( S \) accordingly.
06

- Predict S for Given Values of A and B in 6 Years

Similarly, substitute the predicted values for 6 years from now \( A = 8.9 \) and \( B = 66.25 \) into the equation \( S = a + bA + cB \). Calculate \( S \) accordingly and discuss the validity based on trends and realism compared to previous values.

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.

matrix inversion
Matrix inversion is a key concept for solving systems of linear equations. In the context of linear regression modeling, it helps us find the relationship between dependent and independent variables. When we formulated our equations into a matrix, we converted them into the form \(AX = B\). To solve for the vector \(X\) that contains our coefficients \(a, b, \text{ and } c\), we perform matrix inversion on \(A\) so that \(X = A^{-1}B\). Matrix inversion essentially flips the matrix in a way that when multiplied back by the original matrix, it yields the identity matrix. This method needs the matrix \(A\) to be square (having the same number of equations as unknowns) and non-singular (it must have an inverse). If these conditions are met, you can use this powerful tool to find exact values for your system.
system of linear equations
A system of linear equations is a collection of one or more equations involving the same set of variables. For instance, in our glass sales exercise, we had three linear equations involving variables \(a, b, \text{ and } c\). These equations represent different constraints based on past sales data, and solving them provides a way to predict future values. To solve such a system, we can use various methods including substitution, elimination, or matrix methods. By formulating these equations into a matrix format, solving for coefficients becomes more structured and manageable, especially with the help of matrix inversion.
predictive analytics
Predictive analytics involves using statistical techniques to analyze historical data and make predictions about future outcomes. In the glass sales scenario, we use past sales data to build a linear regression model that predicts future sales based on the number of new building contracts issued and automobiles produced. We find patterns and relationships within the past data through matrix inversion and system of linear equations. Once we have our model, it's used to estimate future values. This method is crucial for businesses as it guides decision-making by forecasting future sales or trends, enabling them to strategize accordingly. For instance, substituting values for future \(A\) and \(B\) into our linear equation helped us predict the sales \(S\) for specific future scenarios.

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

$$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right], \quad B=\left[\begin{array}{ll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right], \quad \text { and } \quad C=\left[\begin{array}{ll} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}\right] $$ where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \((A B) C=A(B C)\) (associative property)

Solve each problem. Purchasing costs The Bread Box, a small neighborhood bakery, sells four main items: sweet rolls, bread, cakes, and pies. The amount of each ingredient (in cups, except for eggs) required for these items is given by matrix \(A\) Eggs lour Sugar Shortening Milk \(\left.\begin{array}{l|ccccc}\text { Rolls (doz) } & 1 & 4 & \frac{1}{4} & \frac{1}{4} & 1 \\ \text { Bread (loaf) } & 0 & 3 & 0 & \frac{1}{4} & 0 \\ \text { Cake } & 4 & 3 & 2 & 1 & 1 \\ \text { Pie (crust) } & 0 & 1 & 0 & \frac{1}{3} & 0\end{array}\right]=A\) The cost (in cents) for each ingredient when purchased in large lots or small lots is given by matrix \(B\) Large Lot Small Lot \(\left.\begin{array}{l|rr}\text { Eggs } & 5 & 5 \\ \text { Flour } & 8 & 10 \\\ \text { Sugar } & 10 & 12 \\ \text { Shortening } & 12 & 15 \\ \text { Milk } & 5 & 6\end{array}\right]=B\) (a) Use matrix multiplication to find a matrix giving the comparative cost per bakery item for the two purchase options. (b) Suppose a day's orders consist of 20 dozen sweet rolls, 200 loaves of bread, 50 cakes, and 60 pies. Write the orders as a \(1 \times 4\) matrix, and, using matrix multiplication, write as a matrix the amount of each ingredient needed to fill the day's orders. (c) Use matrix multiplication to find a matrix giving the costs under the two purchase options to fill the day's orders.

Supply and Demand In many applications of economics, as the price of an item goes up, demand for the item goes down and supply of the item goes up. The price where supply and demand are equal is the equilibrium price, and the resulting sup. ply or demand is the equilibrium supply or equilibrium demand. Suppose the supply of a product is related to its price by the equation $$p=\frac{2}{3} q$$ where \(p\) is in dollars and \(q\) is supply in appropriate units. (Here, \(q\) stands for quantity.) Furthermore, suppose demand and price for the same product are related by $$p=-\frac{1}{3} q+18$$ where \(p\) is price and \(q\) is demand. The system formed by these two equations has solution \((18,12),\) as seen in the graph. (GRAPH CANNOT COPY) Find the demand for the electric can opener at each price. (a) \(\$ 6\) (b) \(\$ 11\) (c) \(\$ 16\)

Supply and Demand In many applications of economics, as the price of an item goes up, demand for the item goes down and supply of the item goes up. The price where supply and demand are equal is the equilibrium price, and the resulting sup. ply or demand is the equilibrium supply or equilibrium demand. Suppose the supply of a product is related to its price by the equation $$p=\frac{2}{3} q$$ where \(p\) is in dollars and \(q\) is supply in appropriate units. (Here, \(q\) stands for quantity.) Furthermore, suppose demand and price for the same product are related by $$p=-\frac{1}{3} q+18$$ where \(p\) is price and \(q\) is demand. The system formed by these two equations has solution \((18,12),\) as seen in the graph. (GRAPH CANNOT COPY) Suppose the demand and price for a certain model of electric can opener are related by \(p=16-\frac{5}{4} q\), where \(p\) is price, in dollars, and \(q\) is demand, in appropriate units. Find the price when the demand is at each level. (a) 0 units (b) 4 units (c) 8 units

Solve each system for \(x\) and y using Cramer's rule. Assume a and b are nonzero constants. $$\begin{array}{l} b^{2} x+a^{2} y=b^{2} \\ a x+b y=a \end{array}$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Probability and 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.