Modeling and Analyzing Real-World Systems in 8th Grade

A farmer has 100 feet of fencing to create a rectangular pen for his sheep. If the length of the pen is represented by x and the width by y, write a system of equations to represent the perimeter. Graph the equations and find the dimensions that maximize the area.

• Length: 25 feet, Width: 25 feet

• Length: 30 feet, Width: 20 feet

• Length: 50 feet, Width: 5 feet

• Length: 40 feet, Width: 10 feet

A school is selling tickets for a concert. Adult tickets cost $10 and student tickets cost $5. If the school wants to raise $500, write a system of equations to represent the number of adult and student tickets sold. Graph the equations and analyze the possible solutions.

• 5x + 10y = 500, x >= 0, y >= 0

• The system of equations is: 10x + 5y = 500, x >= 0, y >= 0.

• 10x + 5y = 250, x >= 0, y >= 0

• 10x + 5y = 1000, x >= 0, y >= 0

Two friends, Alex and Jamie, are saving money for a concert. Alex saves $15 a week, while Jamie saves $10 a week. If Alex starts with $50 and Jamie starts with $30, write a system of equations to model their savings over time. Graph the equations and determine when they will have the same amount saved.

• 5 weeks

• 6 weeks

• 4 weeks

• 7 weeks

A local bakery sells two types of cookies: chocolate chip and oatmeal raisin. The bakery makes a total of 200 cookies each day. If the number of chocolate chip cookies is represented by x and the number of oatmeal raisin cookies by y, write a system of equations to represent the situation. Graph the equations and find the possible combinations of cookies.

• x + 2y = 200, where x >= 0 and y >= 0.

• The system of equations is: x + y = 200, where x >= 0 and y >= 0.

• x - y = 200, where x >= 0 and y >= 0.

• x + y = 100, where x >= 0 and y >= 0.

A car rental company charges a flat fee of $30 plus $0.20 per mile driven. Another company charges a flat fee of $20 plus $0.25 per mile. Write a system of equations to represent the total cost for each company. Graph the equations and determine at what mileage the costs are equal.

• 250 miles

• 150 miles

• 300 miles

• 200 miles

A movie theater has two types of tickets: adult tickets for $12 and child tickets for $8. If the theater sells a total of 150 tickets and makes $1,200, write a system of equations to represent the situation. Graph the equations and analyze the solutions to find the number of adult and child tickets sold.

• 0 adult tickets and 150 child tickets were sold.

• 100 adult tickets and 50 child tickets were sold.

• 50 adult tickets and 100 child tickets were sold.

• 75 adult tickets and 75 child tickets were sold.

A gym offers two membership plans. Plan A costs $30 per month with a one-time fee of $50, while Plan B costs $20 per month with a one-time fee of $100. Write a system of equations to model the total cost over time. Graph the equations and determine after how many months the costs will be the same.

• 5 months

• 10 months

• 7 months

• 3 months

A bookstore sells novels for $15 and textbooks for $25. If the store sold a total of 80 books for $1,500, write a system of equations to represent the situation. Graph the equations and find the number of novels and textbooks sold.

• 30 novels and 50 textbooks

• 40 novels and 40 textbooks

• 50 novels and 30 textbooks

• 60 novels and 20 textbooks

A charity is organizing a fundraiser. They sell cookies for $3 each and brownies for $4 each. If they want to raise $600 and sell a total of 200 items, write a system of equations to represent the situation. Graph the equations and analyze the solutions to find the number of cookies and brownies sold.

• 200 cookies and 0 brownies

• 100 cookies and 100 brownies

• 150 cookies and 50 brownies

• 0 cookies and 200 brownies

A delivery service charges a base fee of $5 plus $2 per mile. Another service charges a base fee of $3 plus $3 per mile. Write a system of equations to represent the total cost for each service. Graph the equations and determine at what distance the costs are equal.

• 2 miles

• 1 mile

• 4 miles

• 3 miles