Algebra 2: Unit 2, Lesson 1 - Practice Problems

Problem 1: Rectangular Schoolyard Fencing

• Given: Schoolyard to be fenced using 150 meters of fencing for three sides, with the school wall as the fourth side.

• Objective: Express the area of the schoolyard as a function of length x.

• Steps:

  • Draw a rectangle with the school wall as one side.

  • Let the two sides be x.

  • The third side: 150 - 2x.

  • Area formula: Area = length * width.

    A(x) = x * (150 - 2x) 
    

• Calculation: Find A(40).

  A(40) = 40 * (150 - 2*40)
         = 40 * 70 
         = 2800 
  

  • Area when x is 40 meters: 2800 meters².

• Domain: Ensure side length remains positive.

  • Solve: 150 - 2x > 0.

    150 > 2x 
    x < 75 
    Domain: 0 < x < 75 
    

Problem 2: Volume of Open-Top Box

• Given: Length x of cutouts from a graph with domain between -1 and 5.
• Objective: Find appropriate domain and maximum volume.
• Steps:
  • Volume must be non-negative.
  • Analyze the graph: Stay within x values where volume > 0.
  • Domain: 0 ≤ x ≤ 2.5.
• Maximum Volume: Highest point on the graph indicates max volume ≈ 15 cubic inches.

Problem 3: Open-Top Box from Square Cardboard

• Given: 10x10 cm cardboard, cutout squares of length x at corners.
• Objective: Write volume as a function of x and find volume when x = 3.
• Steps:
  • Initial cardboard: 10x10 cm.
  • Corners cut: Length x.
  • New dimensions: 10 - 2x by 10 - 2x.
  • Volume formula: Volume = (10 - 2x) * (10 - 2x) * x.

  V(x) = (10 - 2x)² * x
  

• Calculation: Find V(3).

  V(3) = (10 - 2*3)² * 3 
        = 4² * 3 
        = 16 * 3 
        = 48 cm³ 
  

• Volume when x is 3 cm: 48 cm³.

Problem 4: Exponential Growth - Algae on a Pond

• Given: Initial algae coverage is 1/4 square meter, doubling each day.
• Objective: Fill the table showing algae growth.
• Steps:
  • Day 1: 1/4
  • Day 2: 1/2
  • Day 3: 1
  • Day 4: 2
  • Day 5: 4
  • Day 6: 8

Problem 5: Recursive Sequence

• Given: Sequence P with specific values.
• Objective: Define P recursively.
• Steps:

  • Identify pattern: Dividing by 10 each step.

  • Recursive definition:

    P(1) = 5000
    P(n) = P(n-1) / 10, for n ≥ 2
    

Problem 6: Sloth Populations

• Given: Two sloth populations with growth patterns.
• Objective: Describe patterns, extend them, graph, and determine intersection.
• Steps:
  • Population 1: Exponential decay (multiplied by 0.85).
    • Pattern: Continued with 0.85 factor.
  • Population 2: Linear decay (subtracted by 2).
    • Pattern: Continued by subtracting 2.
  • Graphs:
    • Population 1: Exponential curve.
    • Population 2: Linear decrease.
  • Will they intersect?: Approximate equal population around 9 years.