1. Metallurgy
Three iron alloys contain different percents of carbon, chromium, and iron, shown in the matrix below. Alloy W is a type of wrought iron, alloy S is a type of stainless steel, and alloy C is a type of cast iron. Use the outline below to determine how many tons of each alloy can be made with 15 tons of carbon, 19 tons of chromium, and 566 tons of iron.
Alloy Alloy Alloy W S C
Carbon 1% 1% 4% Chromium 0% 5% 3% Iron 99% 94% 93%
1. Let x, y, and z represent the number of tons ins the wrought iron, stainless steel, and cast iron alloys, respectively. Write a linear system to model the situation.
2. Solve the system using matrices.
You can set up a matrix equation, AX=B, and solve the equation for X by finding X=A-1B OR Solve the system using Cramer’s Rule.
3. Express your solutions using a complete sentence. 2. Gold Alloys
Gold jewelry is seldom made of pure gold because pure gold is soft and expensive. Instead, gold is mixed with other metals to produce a harder, less expensive gold alloy. The amount of gold (by weight) in an alloy is measured in karats. Anything made of 24-karat gold is 100% gold. An 18-karat gold mixture is 75% gold.
Three different gold alloys contain the percents of gold, copper, and silver shown in the following matrix. You have 20,144 grams of gold, 766 grams of copper, and 1990 grams of silver. Use the outline below to determine how many grams of each alloy can be made.
Percent by Weight
Alloy Alloy Alloy P Q R
Gold 94% 92% 80% Copper 4% 2% 4% Silver 2% 6% 16%
1. Let x, y, and z represent the total number of grams in alloys P, Q, and R, respectively. Write a linear system to model the situation.
2. Solve the system using matrices.
You can set up a matrix equation, AX=B, and solve the equation for X by finding X=A-1B OR Solve the system using Cramer’s Rule.
3. Express your solutions using a complete sentence. 3. Business
The percent (by age group) of the total amounts spent on three types of shoes in 1996 is shown in the following matrix. The total amounts (in millions) spent by each age group on the three types of shoes were $518.97 (14-17 age group), $336.16 (18-24 age group), and $753.37 (25-34 age group). Answer questions 1, 2, and 3 below to determine how many dollars worth of gym shoes, jogging shoes, and walking shoes were sold in 1996.
Gym Jogging Walking shoes shoes shoes
14-17 0.14 0.13 0.03 18-24 0.05 0.10 0.04 25-34 0.10 0.19 0.11
1. Let x, y, and z represent the amounts of the three types of shoes. Write a linear system to model the situation.
2. Solve the system using matrices.
You can set up a matrix equation, AX=B, and solve the equation for X by finding X=A-1B OR Solve the system using Cramer’s Rule.
3. Express your solutions using a complete sentence. 4. Forestry
A large region of forest has been infected with gypsy moths. The region is roughly triangular, as shown below. From the northernmost vertex A of the region, the distances to the other vertices are 25 miles south and 10 miles east (for vertex B), and 20 miles south and 28 miles east (for vertex C). Answer questions 1, 2, and 3 below to determine the approximate number of square miles in this region.
1. Establish a coordinate system and write the coordinates for points A, B, and C. 2. Write a matrix expression to represent the area of the region. 3. Calculate the result of the matrix expression to find the area.
N A W E
20 25 S
C
B 10
28 5. Real Estate
You own a triangular tract of land, as shown below. To estimate the number of square feet in the tract, you start at one vertex, walk 65 feet east and 50 feet north to the second vertex, and then walk 85 feet west and 30 feet north to the third vertex. Follow items1, 2, and 3 below to determine the approximate area of the tract in square feet.
1. Label the vertices A, B, and C. Establish a coordinate system and write the coordinates for points A, B, and C. 2. Write a matrix expression to represent the area of the region. 3. Calculate the result of the matrix expression to find the area.
85
30
N 50
W E
S 65 6. Investment Portfolio
You have $30,000 to invest in the three stocks shown below. You want an average annual return of 9%. Because Stock X is a high-risk stock, you want the combined investment in Stock Y and Stock Z to be 4 times the amount invested in Stock X. How much should you invest in each type of stock?
Stock Expected Return
Stock X 12% Stock Y 9% Stock Z 8%
1. Let x, y, and z represent the dollar amounts invested in Stocks X, Y, and Z, respectively. Write a linear system to model the situation.
2. Solve the system using matrices.
You can set up a matrix equation, AX=B, and solve the equation for X by finding X=A-1B OR Solve the system using Cramer’s Rule.
3. Express your solutions using a complete sentence.
Y