Functional Relationships

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Answer Key & Explanations

Functional Relationships · Grade 8 · Worksheet 1

1. A rectangular prism has a length of 12 cm, width of 8 cm, and height of 5 cm. If you double all three dimensions to create a larger prism, how many times greater will the volume of the new prism be compared to the original? Answer: 8 Solution: Find the volume of the original prism. The formula for the volume of a rectangular prism is: Volume = length × width × height Original length = 12 cm, width = 8 cm, height = 5 cm So, original volume = 12 × 8 × 5 First, 12 × 8 = 96 Then, 96 × 5 = 480 Original volume = 480 cubic cm.
   Full step-by-step solution

   Step 1: Find the volume of the original prism. The formula for the volume of a rectangular prism is: Volume = length × width × height Original length = 12 cm, width = 8 cm, height = 5 cm So, original volume = 12 × 8 × 5 First, 12 × 8 = 96 Then, 96 × 5 = 480 Original volume = 480 cubic cm. Step 2: Find the dimensions of the new prism after doubling. New length = 12 × 2 = 24 cm New width = 8 × 2 = 16 cm New height = 5 × 2 = 10 cm Step 3: Find the volume of the new prism. New volume = 24 × 16 × 10 First, 24 × 16 = 384 Then, 384 × 10 = 3840 New volume = 3840 cubic cm. Step 4: Compare the new volume to the original volume. Volume ratio = New volume ÷ Original volume Volume ratio = 3840 ÷ 480 3840 ÷ 480 = 8 Step 5: Explain the reasoning. When all three dimensions are doubled, each dimension is multiplied by 2, so the volume is multiplied by 2 × 2 × 2 = 8. This matches our calculation: the new volume is 8 times the original volume. Final answer: 8

2. Emma is analyzing how the height of a plant changes over time. She records the plant's height each week and notices that the relationship between time (in weeks) and height (in centimeters) can be modeled by the equation y = 2x + 5, where x is the number of weeks and y is the height in centimeters. If Emma wants to predict when the plant will reach a height of 25 centimeters, which equation should she solve? Answer: B. 2x + 5 = 25 Solution: The given equation is y = 2x + 5, where y represents the height and x represents the number of weeks. Emma wants to know when the height (y) will be 25 centimeters. Substitute y = 25 into the equation: 25 = 2x + 5.
   Full step-by-step solution

   Step 1: The given equation is y = 2x + 5, where y represents the height and x represents the number of weeks. Step 2: Emma wants to know when the height (y) will be 25 centimeters. Step 3: Substitute y = 25 into the equation: 25 = 2x + 5. Step 4: The equation to solve is 2x + 5 = 25. The correct answer is 2x + 5 = 25.

3. (2.5 × 10³) × (4.0 × 10⁻²) = ? Answer: 100 Solution: Multiply the coefficients: 2.5 × 4.0 = 10.0 Add the exponents: 3 + (-2) = 1 Combine the results: 10.0 × 10¹ Convert to standard form: 10.0 × 10 = 100 The answer is 100.
   Full step-by-step solution

   Step 1: Multiply the coefficients: 2.5 × 4.0 = 10.0 Step 2: Add the exponents: 3 + (-2) = 1 Step 3: Combine the results: 10.0 × 10¹ Step 4: Convert to standard form: 10.0 × 10 = 100 The answer is 100.

4. Isabella is training for a charity run. She records her distance (in miles) over time (in minutes) during a long training session. For the first 15 minutes, she runs at a steady pace, covering 0.1 miles each minute. Then, she slows down and covers only 0.05 miles each minute for the next 20 minutes. Finally, she walks the last 10 minutes, covering a constant 0.02 miles each minute. Describe how the total distance she has covered changes over time during each of the three intervals. Is the relationship increasing, decreasing, or constant in each interval? Answer: During the first 15 minutes, the total distance is increasing at a constant rate. During the next 20 minutes, the total distance is still increasing, but at a slower constant rate. During the last 10 minutes, the total distance continues to increase, but at an even slower constant rate. Solution: Isabella is tracking total distance over time. As time passes, she keeps moving forward, so her total distance always increases (never decreases or stays flat). First interval (0 to 15 minutes): She runs at 0.1 miles per minute.
   Full step-by-step solution

   Step 1: Understand the context. Isabella is tracking total distance over time. As time passes, she keeps moving forward, so her total distance always increases (never decreases or stays flat). Step 2: First interval (0 to 15 minutes): She runs at 0.1 miles per minute. This is a constant positive rate, so total distance increases steadily (linear increase). The relationship is increasing and constant rate. Step 3: Second interval (15 to 35 minutes): She runs at 0.05 miles per minute, which is still positive but slower. Total distance continues to increase, but the rate of increase is slower. The relationship is still increasing, but at a slower constant rate. Step 4: Third interval (35 to 45 minutes): She walks at 0.02 miles per minute, still positive but even slower. Total distance keeps increasing, but at the slowest constant rate so far. Conclusion: In all three intervals, the total distance is increasing (never decreasing or constant), but the rate of increase changes from faster to slower. The answer describes each interval's behavior as increasing with a constant rate (different for each).

5. Mason is tracking the water level in a cylindrical tank over the course of a week. He records the height of water (in inches) each day. On Monday, the height is 27 inches. On Tuesday, it increases by 2 inches. On Wednesday, it stays the same. On Thursday, it decreases by 7 inches. On Friday, it decreases by another 2 inches. On Saturday, it increases by 12 inches. Describe qualitatively how the height of the water changes over the week. Be sure to identify each interval (day to day) as increasing, decreasing, or constant. Answer: Monday to Tuesday: increasing; Tuesday to Wednesday: constant; Wednesday to Thursday: decreasing; Thursday to Friday: decreasing; Friday to Saturday: increasing. Solution: Monday height = 27 inches. Tuesday height = 27 + 2 = 29 inches. Since 29 > 27, the function is increasing from Monday to Tuesday.
   Full step-by-step solution

   Step 1: Monday height = 27 inches. Tuesday height = 27 + 2 = 29 inches. Since 29 > 27, the function is increasing from Monday to Tuesday. Step 2: Tuesday height = 29 inches. Wednesday height = 29 inches (stays same). Since it did not change, the function is constant from Tuesday to Wednesday. Step 3: Wednesday height = 29 inches. Thursday height = 29 - 7 = 22 inches. Since 22 < 29, the function is decreasing from Wednesday to Thursday. Step 4: Thursday height = 22 inches. Friday height = 22 - 2 = 20 inches. Since 20 < 22, the function is decreasing from Thursday to Friday. Step 5: Friday height = 20 inches. Saturday height = 20 + 12 = 32 inches. Since 32 > 20, the function is increasing from Friday to Saturday. The answer is: Monday to Tuesday: increasing; Tuesday to Wednesday: constant; Wednesday to Thursday: decreasing; Thursday to Friday: decreasing; Friday to Saturday: increasing.