Functional Relationships

lessonbunny.com

Answer Key & Explanations

Functional Relationships · Grade 8 · Worksheet 3

1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A second triangle is created by rotating the first triangle 90 degrees counterclockwise about the origin. What are the coordinates of the vertex that was originally at (6,8) after this rotation? Answer: (-8,6) Solution: A = (0, 0) B = (6, 0) C = (6, 8) We are rotating the triangle 90° counterclockwise about the origin (0, 0). We need the new coordinates of the vertex that was originally at (6, 8) after rotation.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a right triangle with vertices: A = (0, 0) B = (6, 0) C = (6, 8) We are rotating the triangle 90° counterclockwise about the origin (0, 0). We need the new coordinates of the vertex that was originally at (6, 8) after rotation. --- **Step 2: Recall the rotation rule** For a point (x, y) rotated 90° counterclockwise about the origin: New coordinates = (-y, x) This is a standard rule in coordinate geometry. --- **Step 3: Apply the rule to point (6, 8)** Original point: x = 6, y = 8 After rotation: x' = -y = -8 y' = x = 6 So new coordinates = (-8, 6) --- **Step 4: Verify with reasoning** When rotating 90° counterclockwise: - The point (6, 8) is in the first quadrant. - After rotation, it should end up in the second quadrant (negative x, positive y). - The distance from the origin should remain the same. Distance before rotation: sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 Distance after rotation: sqrt((-8)^2 + 6^2) = sqrt(64 + 36) = sqrt(100) = 10 ✓ --- **Final Answer:** (-8, 6)

2. Liam is filling a large cylindrical fish tank with water. He starts with the tank empty and turns on a hose that fills the tank at a constant rate of 3 gallons per minute. After 7 minutes, he notices the tank is 21 gallons full. He then turns off the hose and lets the fish tank sit for 5 minutes. During this time, no water is added or removed. After the 5-minute rest, Liam turns the hose back on at the same rate of 3 gallons per minute for another 11 minutes. Describe qualitatively how the volume of water in the tank changes over the entire 23-minute period. In your description, identify each interval where the volume is increasing, decreasing, or constant, and explain what is happening in each interval. Answer: The volume is increasing from 0 to 7 minutes, constant from 7 to 12 minutes, and increasing from 12 to 23 minutes. Solution: Identify the three time intervals: 0 to 7 minutes (hose on), 7 to 12 minutes (hose off), 12 to 23 minutes (hose on again).
   Full step-by-step solution

   Step 1: Identify the three time intervals: 0 to 7 minutes (hose on), 7 to 12 minutes (hose off), 12 to 23 minutes (hose on again). Step 2: For the first interval (0 to 7 minutes), the hose is adding water at a constant rate, so the volume is increasing. Step 3: For the second interval (7 to 12 minutes), the hose is off and no water is added or removed, so the volume is constant. Step 4: For the third interval (12 to 23 minutes), the hose is on again, so the volume is increasing. Step 5: Therefore, the volume increases from 0 to 7 minutes, stays constant from 7 to 12 minutes, and increases again from 12 to 23 minutes.

3. Sophia is training for a charity walkathon. She records the distance she walks each day over a two-week period. During the first week (days 1 through 7), she walks 2 miles every day. In the second week (days 8 through 14), she increases her distance by 0.5 miles each day, starting at 2.5 miles on day 8 and ending at 5.5 miles on day 14. After day 14, she feels tired and walks only 3 miles each day for the next three days (days 15, 16, and 17). Describe how the distance Sophia walks each day changes over the 17 days. In your description, identify the intervals where the distance is increasing, decreasing, or constant. Answer: From day 1 to day 7, the distance is constant at 2 miles per day. From day 7 to day 14, the distance is increasing by 0.5 miles each day. From day 14 to day 15, the distance decreases from 5.5 miles to 3 miles. From day 15 to day 17, the distance is constant at 3 miles per day. Solution: Analyze the first week (days 1–7). Sophia walks 2 miles every day. The distance does not change from day to day, so it is constant.
   Full step-by-step solution

   Step 1: Analyze the first week (days 1–7). Sophia walks 2 miles every day. The distance does not change from day to day, so it is constant. Step 2: Analyze the second week (days 8–14). Sophia increases her distance by 0.5 miles each day, starting at 2.5 miles on day 8 and ending at 5.5 miles on day 14. The distance goes up each day, so it is increasing. Step 3: Analyze days 14–15. On day 14, Sophia walks 5.5 miles. On day 15, she walks only 3 miles. The distance drops from 5.5 to 3, so it is decreasing. Step 4: Analyze days 15–17. Sophia walks 3 miles each day on days 15, 16, and 17. The distance does not change, so it is constant again. Final answer: The distance is constant from day 1 to day 7, increasing from day 7 to day 14, decreasing from day 14 to day 15, and constant from day 15 to day 17.

4. Maya is comparing two cell phone plans. Plan A has a $15 monthly fee plus $0.10 per minute of talk time. Plan B has a $25 monthly fee plus $0.05 per minute. Maya wants to know at what number of minutes used the total monthly cost would be the same for both plans. How many minutes would that be? Answer: 200 Solution: Write an expression for Plan A's total cost: 15 + 0.10x, where x is minutes used. Write an expression for Plan B's total cost: 25 + 0.05x.
   Full step-by-step solution

   Step 1: Write an expression for Plan A's total cost: 15 + 0.10x, where x is minutes used. Step 2: Write an expression for Plan B's total cost: 25 + 0.05x. Step 3: Set the expressions equal to find when costs are the same: 15 + 0.10x = 25 + 0.05x. Step 4: Subtract 15 from both sides: 0.10x = 10 + 0.05x. Step 5: Subtract 0.05x from both sides: 0.05x = 10. Step 6: Divide both sides by 0.05: x = 10 ÷ 0.05 = 200. The answer is 200 minutes.

5. A rocket travels at a constant speed of 2.5 × 10^4 kilometers per hour. How many hours will it take for the rocket to travel a distance of 3.75 × 10^5 kilometers? Answer: 15 Solution: Write the formula for time: time = distance ÷ speed Substitute the given values: time = (3.75 × 10^5) ÷ (2.5 × 10^4) Divide the coefficients: 3.75 ÷ 2.5 = 1.5 Divide the powers of 10: 10^5 ÷ 10^4 = 10^(5-4) = 10^1 Combine the results: 1.5 × 10^1 Convert to standard form: 1.5 × 10 = 15 The answer…
   Full step-by-step solution

   Step 1: Write the formula for time: time = distance ÷ speed Step 2: Substitute the given values: time = (3.75 × 10^5) ÷ (2.5 × 10^4) Step 3: Divide the coefficients: 3.75 ÷ 2.5 = 1.5 Step 4: Divide the powers of 10: 10^5 ÷ 10^4 = 10^(5-4) = 10^1 Step 5: Combine the results: 1.5 × 10^1 Step 6: Convert to standard form: 1.5 × 10 = 15 The answer is 15.

6. A right triangle is drawn on a coordinate plane with vertices at (0,0), (8,0), and (8,6). A second triangle is created by reflecting the first triangle across the y-axis. What are the coordinates of the vertex that was originally at (8,6) after this reflection? Answer: (-8,6) Solution: The original vertex is at (8,6) Reflection across the y-axis means we change the sign of the x-coordinate while keeping the y-coordinate the same The x-coordinate 8 becomes -8 The y-coordinate 6 stays as 6 Therefore, the reflected point is (-8,6) The answer is (-8,6).
   Full step-by-step solution

   Step 1: The original vertex is at (8,6) Step 2: Reflection across the y-axis means we change the sign of the x-coordinate while keeping the y-coordinate the same Step 3: The x-coordinate 8 becomes -8 Step 4: The y-coordinate 6 stays as 6 Step 5: Therefore, the reflected point is (-8,6) The answer is (-8,6).