Rate of Change

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Rate of Change · Grade 8 · Worksheet 1

1. Maya is tracking the growth of her bamboo plant. She notices it grows at a constant rate each week. After 4 weeks, the plant is 2.8 meters tall, and after 8 weeks, it reaches 4.4 meters tall. What was the initial height of the bamboo plant when Maya started tracking it? Answer: 1.2 Solution: Identify the two data points: (4, 2.8) and (8, 4.4), where x is weeks and y is height in meters. Calculate the growth rate per week: (4.4 - 2.8) / (8 - 4) = 1.6 / 4 = 0.4 meters per week.
   Full step-by-step solution

   Step 1: Identify the two data points: (4, 2.8) and (8, 4.4), where x is weeks and y is height in meters. Step 2: Calculate the growth rate per week: (4.4 - 2.8) / (8 - 4) = 1.6 / 4 = 0.4 meters per week. Step 3: Work backwards to find the initial height. After 4 weeks, the height is 2.8 meters. The plant grew 4 weeks × 0.4 meters/week = 1.6 meters in that time. Step 4: Subtract the growth from the week 4 height to find the initial height: 2.8 - 1.6 = 1.2 meters. The initial height was 1.2 meters.

2. (4.2 × 10⁵) ÷ (6 × 10²) = ? Answer: 700 Solution: Separate the coefficients and exponents: (4.2 ÷ 6) × (10⁵ ÷ 10²) Divide the coefficients: 4.2 ÷ 6 = 0.7 Divide the powers of 10: 10⁵ ÷ 10² = 10^(5-2) = 10³ Multiply the results: 0.7 × 10³ Convert to standard form: 0.7 × 1000 = 700 The answer is 700.
   Full step-by-step solution

   Step 1: Separate the coefficients and exponents: (4.2 ÷ 6) × (10⁵ ÷ 10²) Step 2: Divide the coefficients: 4.2 ÷ 6 = 0.7 Step 3: Divide the powers of 10: 10⁵ ÷ 10² = 10^(5-2) = 10³ Step 4: Multiply the results: 0.7 × 10³ Step 5: Convert to standard form: 0.7 × 1000 = 700 The answer is 700.

3. Liam is tracking the growth of his sunflower plant. After 2 weeks, the plant was 8 inches tall. After 6 weeks, it had grown to 20 inches tall. If the plant continues to grow at the same constant rate, what was its initial height when Liam first started measuring? Answer: 2 inches Solution: Let’s go step-by-step. - At week 2, height = 8 inches - At week 6, height = 20 inches - Need initial height at week 0 From week 2 to week 6, that’s 6 − 2 = 4 weeks. Height change = 20 − 8 = 12 inches.
   Full step-by-step solution

   Let’s go step-by-step. --- **Step 1: Understand the problem** We know: - At week 2, height = 8 inches - At week 6, height = 20 inches - Growth rate is constant - Need initial height at week 0 --- **Step 2: Find the growth rate** From week 2 to week 6, that’s 6 − 2 = 4 weeks. Height change = 20 − 8 = 12 inches. So in 4 weeks, it grew 12 inches. Growth rate per week = 12 ÷ 4 = 3 inches/week. --- **Step 3: Work backwards to week 0** At week 2, height = 8 inches. Since it grows 3 inches per week, the height 2 weeks earlier (week 0) would be: Height at week 0 = 8 − (3 × 2) = 8 − 6 = 2 inches. --- **Step 4: Check with week 6 data** From week 0 to week 6: Height = initial + (3 × 6) = 2 + 18 = 20 inches. Matches the given data. --- **Final answer:** The initial height was **2 inches**.

4. Liam is tracking the growth of his sunflower plant. In the first week, it grew 8 cm. Each subsequent week, it grows 75% of the amount it grew the previous week. What is the total height the sunflower will eventually reach from its starting point, considering this pattern continues indefinitely? Answer: 32 Solution: Let’s go step-by-step. - Week 1 growth: 8 cm - Week 2 growth: 8 × 0.75 - Week 3 growth: (8 × 0.75) × 0.75 = 8 × (0.75)^2 - And so on.
   Full step-by-step solution

   Let’s go step-by-step. --- **Step 1: Understand the problem** - Week 1 growth: 8 cm - Week 2 growth: 8 × 0.75 - Week 3 growth: (8 × 0.75) × 0.75 = 8 × (0.75)^2 - And so on. So the growth each week forms a geometric sequence: 8, 8 × 0.75, 8 × (0.75)^2, 8 × (0.75)^3, ... --- **Step 2: Identify the total growth** The total eventual height is the sum of all these weekly growths: Total = 8 + 8 × 0.75 + 8 × (0.75)^2 + 8 × (0.75)^3 + ... This is an infinite geometric series. --- **Step 3: Recall the infinite geometric series formula** Sum = a / (1 - r) where a = first term = 8 r = common ratio = 0.75 --- **Step 4: Apply the formula** Total = 8 / (1 - 0.75) = 8 / 0.25 = 8 × 4 = 32 --- **Step 5: Interpret the result** The total eventual height from the starting point is 32 cm. --- **Final answer:** 32

5. Noah is looking at a graph that shows the total amount of water in a tank over time as it is being filled. The graph is a straight line. The line passes through the point (0, 16) and the point (6, 76). What does the slope of this line represent in the context of filling the tank, and what is its value? Answer: The slope represents the rate at which water is being added to the tank, and its value is 10 liters per minute. Solution: Identify the two points on the line: (0, 16) and (6, 76). The point (0, 16) means that at time 0 minutes, the tank already has 16 liters of water.
   Full step-by-step solution

   Step 1: Identify the two points on the line: (0, 16) and (6, 76). Step 2: The point (0, 16) means that at time 0 minutes, the tank already has 16 liters of water. This is the initial value (y-intercept), representing water already in the tank before filling starts. Step 3: To find the slope (rate of change), use the formula: slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1). Step 4: Substitute the values: slope = (76 - 16) / (6 - 0) = 60 / 6 = 10. Step 5: The slope is 10. Since the y-axis is total water (in liters) and the x-axis is time (in minutes), the slope represents the rate of water being added per minute. Step 6: Interpretation: Water is being added to the tank at a rate of 10 liters per minute, starting from an initial 16 liters. The answer is: The slope represents the rate at which water is being added to the tank, and its value is 10 liters per minute.

6. A linear function has a slope of 15 and passes through the point (0, 45). What is the initial value of the function? Answer: 45 Solution: The initial value of a linear function is the y-intercept, which is the value of y when x = 0. The problem states the function passes through the point (0, 45). At x = 0, the y-coordinate is 45.
   Full step-by-step solution

   Step 1: The initial value of a linear function is the y-intercept, which is the value of y when x = 0. Step 2: The problem states the function passes through the point (0, 45). Step 3: At x = 0, the y-coordinate is 45. Step 4: Therefore, the initial value is 45. The answer is 45.

7. 2³ × 3² - √144 = ? Answer: 60 Solution: Calculate the exponents first: 2³ = 8 and 3² = 9 Calculate the square root: √144 = 12 Perform the multiplication: 8 × 9 = 72 Perform the subtraction: 72 - 12 = 60 The answer is 60.
   Full step-by-step solution

   Step 1: Calculate the exponents first: 2³ = 8 and 3² = 9 Step 2: Calculate the square root: √144 = 12 Step 3: Perform the multiplication: 8 × 9 = 72 Step 4: Perform the subtraction: 72 - 12 = 60 The answer is 60.

8. Amelia is saving money to buy a new video game. Amelia already has $35 saved up. Each week, Amelia earns $48 from doing chores and adds it to the savings. Assuming Amelia saves at a constant rate, identify the rate of change (weekly savings) and the initial value (starting amount). Answer: 48 Solution: The rate of change is the amount saved each week: $48 per week. The initial value is the amount already saved: $35.
   Full step-by-step solution

   The rate of change is the amount saved each week: $48 per week. The initial value is the amount already saved: $35. So the rate of change is 48 and the initial value is 35.