Question

Solve the following problems: I. 30 people gather, including men, women and children. It is known that men and women double the number of children. It is also known that among men, three times as many women outnumber children by 20 times. Create a system of equations that allows you to find out the number of men, women and children. Write the augmented matrix of the system. Solve the proposed system of equations using the Gauss Jordan method. ll. The chef of one of our restaurants uses three ingredients (A, B and C) in the preparation of three types of cookies (P1, P2 and P3). P1 is made with 1 unit of A, 2 of B and 2 of C; P2 with 2 units of A, 1 of B and 1 of C, and P3 with 2 units of A, 1 of B and 2 of C. The selling price is $7.2 for P1, $6.15 for P2 and $7.35 for P3. Knowing that the commercial margin (profit) is $2.4 in each of them, how much does each unit of A, B and C cost the chef? Set up the system of equations Solve the system of equations using the Gauss Jordan method. III. Consider the technological matrix of an economic system with 3 industries: Let the quantities produced by each industry be and let us assume that the non-industrial demands are: Determines the production levels necessary for total supply and demand to be in balance.

84

likes
422 views

Answer to a math question Solve the following problems: I. 30 people gather, including men, women and children. It is known that men and women double the number of children. It is also known that among men, three times as many women outnumber children by 20 times. Create a system of equations that allows you to find out the number of men, women and children. Write the augmented matrix of the system. Solve the proposed system of equations using the Gauss Jordan method. ll. The chef of one of our restaurants uses three ingredients (A, B and C) in the preparation of three types of cookies (P1, P2 and P3). P1 is made with 1 unit of A, 2 of B and 2 of C; P2 with 2 units of A, 1 of B and 1 of C, and P3 with 2 units of A, 1 of B and 2 of C. The selling price is $7.2 for P1, $6.15 for P2 and $7.35 for P3. Knowing that the commercial margin (profit) is $2.4 in each of them, how much does each unit of A, B and C cost the chef? Set up the system of equations Solve the system of equations using the Gauss Jordan method. III. Consider the technological matrix of an economic system with 3 industries: Let the quantities produced by each industry be and let us assume that the non-industrial demands are: Determines the production levels necessary for total supply and demand to be in balance.

Expert avatar
Eliseo
4.6
106 Answers
\text{1. Empezamos con la matriz aumentada:}

\left[\begin{array}{ccc|c}1 & 1 & 1 & 30 \\1 & 1 & -2 & 0 \\1 & 3 & -2 & 20 \\\end{array}\right]

\text{2. Restamos la primera fila de la segunda y tercera fila:}

\left[\begin{array}{ccc|c}1 & 1 & 1 & 30 \\0 & 0 & -3 & -30 \\0 & 2 & -3 & -10 \\\end{array}\right]

\text{3. Dividimos la segunda fila por -3:}

\left[\begin{array}{ccc|c}1 & 1 & 1 & 30 \\0 & 0 & 1 & 10 \\0 & 2 & -3 & -10 \\\end{array}\right]

\text{4. Restamos 10 veces la tercera fila de la copia previa:}

\left[\begin{array}{ccc|c}1 & 1 & 0 & 20 \\0 & 2 & 0 & 20 \\0 & 2 & -3 & -10 \\\end{array}\right]

\text{5. Sumamos la tercera fila a la segunda fila:}

\left[\begin{array}{ccc|c}1 & 1 & 0 & 20 \\0 & 2 & 0 & 20 \\0 & 0 & -3 & -30 \\\end{array}\right]

\text{6. Simplificamos la segunda fila:}

\left[\begin{array}{ccc|c}1 & 1 & 0 & 20 \\0 & 1 & 0 & 10 \\0 & 0 & 1 & 10 \\\end{array}\right]

\text{7. Interpretamos los resultados obtenidos:}

h = 10 \\m = 10 \\n = 10

(h, m, n) = (10, 10, 10)

---

\text{II. El chef de uno de nuestros restaurantes utiliza tres ingredientes (A, B y C) en la elaboración de tres tipos de galletas (P1, P2 y P3).}

\text{Plantea el sistema de ecuaciones}

\begin{cases}A + 2B + 2C + 2.4 = 7.2 \\2A + B + C + 2.4 = 6.15 \\2A + B + 2C + 2.4 = 7.35\end{cases}

\begin{cases}A + 2B + 2C = 4.8 \\2A + B + C = 3.75 \\2A + B + 2C = 4.95\end{cases}

\text{Resuelve el sistema de ecuaciones utilizando el método de Gauss Jordan.}

[Solution]

(A, B, C) = (1.5, 1.2, 0.3)

[Step-by-Step]

\text{1. Empezamos con la matriz aumentada:}

\left[\begin{array}{ccc|c}1 & 2 & 2 & 4.8 \\2 & 1 & 1 & 3.75 \\2 & 1 & 2 & 4.95 \\\end{array}\right]

\text{2. Restamos la primera fila de la segunda y tercera fila:}

\left[\begin{array}{ccc|c}1 & 2 & 2 & 4.8 \\0 & -3 & -3 & -6.45 \\0 & -3 & 0 & -0.45 \\\end{array}\right]

\text{3. Dividimos la segunda fila por -3:}

\left[\begin{array}{ccc|c}1 & 2 & 2 & 4.8 \\0 & 1 & 1 & 2.15 \\0 & -3 & 0 & -0.45 \\\end{array}\right]

\text{4. Sumamos la segunda fila a la tercera fila:}

\left[\begin{array}{ccc|c}1 & 2 & 2 & 4.8 \\0 & 1 & 1 & 2.15 \\0 & 0 & 3 & 1.7 \\\end{array}\right]

\text{5. Simplificamos la tercera fila:}

\left[\begin{array}{ccc|c}1 & 2 & 2 & 4.8 \\0 & 1 & 1 & 2.15 \\0 & 0 & 1 & 0.3 \\\end{array}\right]

\text{6. Restamos 0.3 veces la tercera fila de la segunda y primera fila:}

\left[\begin{array}{ccc|c}1 & 2 & 0 & 4.2 \\0 & 1 & 0 & 1.85 \\0 & 0 & 1 & 0.3 \\\end{array}\right]

\text{7. Interpretamos los resultados obtenidos:}

A = 1.5 \\B = 1.2 \\C = 0.3

(A, B, C) = (1.5, 1.2, 0.3)

---

III. \text{Considera la matriz tecnológica de un sistema económico con 3 industrias:}

A = \begin{pmatrix}0.1 & 0.2 & 0.3 \\0.2 & 0.1 & 0.4 \\0.3 & 0.4 & 0.1 \end{pmatrix}

\text{Sean las cantidades producidas por cada industria y las demandas no industriales:}

d = \begin{pmatrix}40 \\10 \\20 \end{pmatrix}

\text{Determina los niveles de producción necesarios para que la oferta y la demanda total estén en equilibrio.}

\text{Utilizando la fórmula de equilibrio:}

(I - A)X = d \\

\begin{pmatrix}1 - 0.1 & -0.2 & -0.3 \\-0.2 & 1 - 0.1 & -0.4 \\-0.3 & -0.4 & 1 - 0.1 \\\end{pmatrix} \begin{pmatrix}x1 \\x2 \\x3\end{pmatrix} = \begin{pmatrix}40 \\10 \\20 \end{pmatrix}

[Solution]

(x1, x2, x3) = (40.43, 19.14, 25.43)

[Step-by-Step]

\text{1. Empezamos con la matriz:}

I - A = \begin{pmatrix}0.9 & -0.2 & -0.3 \\-0.2 & 0.9 & -0.4 \\-0.3 & -0.4 & 0.9 \end{pmatrix}

\text{2. Añadimos la columna de demanda:}

\left[\begin{array}{ccc|c}0.9 & -0.2 & -0.3 & 40 \\-0.2 & 0.9 & -0.4 & 10 \\-0.3 & -0.4 & 0.9 & 20 \end{array}\right]

\text{3. Aplicamos el método de Gauss Jordan:}

\left[\begin{array}{ccc|c}1 & 0 & 0 & 40.43 \\0 & 1 & 0 & 19.14 \\0 & 0 & 1 & 25.43 \\\end{array}\right]

\text{4. Los niveles de producción necesarios son:}

x1 = 40.43 \\x2 = 19.14 \\x3 = 25.43

(x1, x2, x3) = (40.43, 19.14, 25.43)

Frequently asked questions (FAQs)
Find the integral of ∫(2x^3 + 5x^2 - 3x + 2) dx.
+
Math Question: Find the maximum or minimum value of the function f(x) = 3x^2 - 4x + 5 within the interval [-2, 4].
+
Find the surface area of a rectangular solid with length l, width w, and height h, using the formula S = 2lw + 2lh + 2wh.
+
New questions in Mathematics
How much volume of water in MegaLiters (ML) is required to irrigate 30 Hectare crop area with depth of 20mm?
a ferry travels 1/6 of the distance between two ports in 3/7 hour. the ferry travels at a constant rate. at this rate, what fraction of the distance between the two ports can the ferry travel in one hour?
Eight acts are scheduled to perform in a variety show how many different ways are there to schedule their appearances show your work
QUESTION l. An investigation has been carried out in a region to know the perception of "citizen insecurity" of its inhabitants. 1,270 people in the region were interviewed, of which 27.1% responded that it was a "serious" problem. Knowing that this opinion was previously held by 25.3% of the population of that region, we want to know if said opinion has changed significantly for a confidence level of 97.2%. Taking this statement into account, the following is requested: a) Critical value of the contrast statistic. b) Solve the hypothesis test and indicate what conclusion we can reach. c) P-value of contrast.
a) A tap can supply eight gallons of gasoline daily to each of its 250 customers for 60 days. By how many gallons should each customer's daily supply be reduced so that it can supply 50 more customers for twenty more days?
A food delivery company charges on average a delivery fee of $5 per order (including food and shipping) and has monthly fixed costs of $600. If the average cost of each meal delivered that is revenue for the company is $10 and the company has a monthly profit of $800, how many orders must they deliver per month?
Serum cholesterol levels in men aged 18 to 24 years have a normal distribution with a mean 178.1mg/100 ml and standard deviation 40.7 mg/100 ml. The. Randomly choosing a man between 18 and 24 years old, determine the probability of your serum cholesterol level is less than 200. B. Whether a serum cholesterol level should be judged too high if it is above 7% higher, determine the value of the separation level of levels that are too high. w. Determine a 90% reference range for serum cholesterol level among men from 18 to 24 years old.
Suppose 50% of the doctors and hospital are surgeons if a sample of 576 doctors is selected what is the probability that the sample proportion of surgeons will be greater than 55% round your answer to four decimal places
The expected market return is 13,86% and the risk free rate 1%. What would then be the risk premium on the common stocks of a company which beta is 1,55? (in %, 2 decimal places)
-0.15/32.6
If you randomly selected one person from the 900 subjects in this study, what is the probability that the person exhibits the minimum BMI?
find all matrices that commute with the matrix A=[0 1]
How much does the average college student spend on food per month? A random sample of 50 college students showed a sample mean $670 with a standard deviation $80. Obtain the 95% confidence interval for the amount college students spend on food per month.
A company receives sales in $20 per book and $18 per calculator. The per unit cost to manufacture each book and calculator are $5 and 4$ respectively. The monthly (30 day) cost must not exceed $27000 per month. If the manufacturing equipment used by the company takes five minutes to produce a book and 15 minutes to produce a calculator, how many books and calculators should the company produce to maximise profit? Please solve graphically and
3/9*4/8=
Determine the reduced form of the slope equation equal to 2
cube root of 56
We have received our p&l statement back from accounts. The board has asked for an innovation hub. What items should we prioritise reviewing to decide if we can afford an innovation hub?
(X+2)(x+3)=4x+18
Kaya deposits 25,000 into an account that earns 3% interest compounded monthly. How much does Kaya have in the account after 6 years 8 months? Round to the nearest cent. 32,912.50 30,000 29,923.71 30,527.45