Rate of Change

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Answer Key & Explanations

Rate of Change · Grade 8 · Worksheet 2

1. (4.2 × 10⁵) × (2.5 × 10³) = ? Answer: 1.05 × 10⁹ Solution: Multiply the coefficients: 4.2 × 2.5 = 10.5 Add the exponents: 5 + 3 = 8 Combine results: 10.5 × 10⁸ Convert to proper scientific notation: 10.5 × 10⁸ = 1.05 × 10¹ × 10⁸ = 1.05 × 10⁹ The answer is 1.05 × 10⁹.
   Full step-by-step solution

   Step 1: Multiply the coefficients: 4.2 × 2.5 = 10.5 Step 2: Add the exponents: 5 + 3 = 8 Step 3: Combine results: 10.5 × 10⁸ Step 4: Convert to proper scientific notation: 10.5 × 10⁸ = 1.05 × 10¹ × 10⁸ = 1.05 × 10⁹ The answer is 1.05 × 10⁹.

2. A linear function has a slope of 6 and passes through the point (0, 11). What is the initial value of the function? Answer: 11 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, 11). At x = 0, the y-coordinate is 11.
   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, 11). Step 3: At x = 0, the y-coordinate is 11. Step 4: Therefore, the initial value is 11. The answer is 11.

3. A linear function models the amount of water in a tank. The function has a slope of 7 and passes through the point (0, 13). What is the initial value of the function? Answer: 13 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, 13). At x = 0, the y-coordinate is 13.
   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, 13). Step 3: At x = 0, the y-coordinate is 13. Step 4: Therefore, the initial value is 13. The answer is 13.

4. Liam is tracking the growth of a bamboo plant in his garden. He notices that the plant's height increases at a constant rate each week. After 3 weeks, the bamboo is 42 inches tall, and after 7 weeks, it is 82 inches tall. What was the initial height of the bamboo plant when Liam started tracking it? Answer: 12 inches Solution: Let’s go step by step. The bamboo grows at a constant rate each week. - After 3 weeks → height = 42 inches - After 7 weeks → height = 82 inches We want the initial height (at week 0).
   Full step-by-step solution

   Let’s go step by step. --- **Step 1: Understand the problem** The bamboo grows at a constant rate each week. We know: - After 3 weeks → height = 42 inches - After 7 weeks → height = 82 inches We want the initial height (at week 0). --- **Step 2: Find the growth rate per week** From week 3 to week 7 is 4 weeks. Height increased from 42 to 82 inches. Increase = 82 − 42 = 40 inches over 4 weeks. So growth rate per week = 40 / 4 = 10 inches per week. --- **Step 3: Work backwards to initial height** At week 3, height = 42 inches. Since it grows 10 inches per week, the height 3 weeks earlier (at week 0) is: Height at week 0 = 42 − (10 × 3) = 42 − 30 = 12 inches. --- **Step 4: Check with week 7 data** At week 7, height = 82 inches. Height at week 0 = 82 − (10 × 7) = 82 − 70 = 12 inches. Both match. --- **Final answer:** The initial height was **12 inches**.

5. Tom is saving money to buy a new video game. Tom already has $22 saved up. Each week, Tom earns $41 from doing chores and adds it to the savings. Assuming Tom saves at a constant rate, identify the rate of change (weekly savings) and the initial value (starting amount). Answer: 41 Solution: The rate of change is the amount saved each week: $41 per week. The initial value is the amount already saved: $22.
   Full step-by-step solution

   The rate of change is the amount saved each week: $41 per week. The initial value is the amount already saved: $22. So the rate of change is 41 and the initial value is 22.

6. Liam is tracking the growth of a bamboo plant in his garden. The plant started at 15 cm tall and has been growing at a constant rate of 8 cm per week. Meanwhile, his sunflower started at 40 cm tall and has been growing at a constant rate of 3 cm per week. After how many weeks will the bamboo and sunflower be the same height? Answer: 5 Solution: Let’s define the height of each plant after a certain number of weeks. Write the height equations. Let x = number of weeks from the start.
   Full step-by-step solution

   Let’s define the height of each plant after a certain number of weeks. Step 1: Write the height equations. Let x = number of weeks from the start. Bamboo: Initial height = 15 cm Growth rate = 8 cm/week Height after x weeks = 15 + 8x Sunflower: Initial height = 40 cm Growth rate = 3 cm/week Height after x weeks = 40 + 3x Step 2: Set the heights equal to find when they are the same. 15 + 8x = 40 + 3x Step 3: Solve for x. Subtract 3x from both sides: 15 + 8x - 3x = 40 + 3x - 3x 15 + 5x = 40 Subtract 15 from both sides: 15 + 5x - 15 = 40 - 15 5x = 25 Divide both sides by 5: x = 25 / 5 x = 5 Step 4: Interpret the result. After 5 weeks, both plants will be the same height. Check: Bamboo: 15 + 8*5 = 15 + 40 = 55 cm Sunflower: 40 + 3*5 = 40 + 15 = 55 cm Yes, both are 55 cm after 5 weeks. Answer: 5

7. A scientist is studying bacterial growth in a lab. The initial population is 800 bacteria, and the population doubles every 3 hours. Write an exponential function in the form y = a(b)^x that models the bacterial population, where x is the number of hours that have passed. What is the value of b in this function? Answer: 2^(1/3) Solution: Initial population = 800 bacteria. Doubling time = 3 hours. y = a * (b)^x where x is in hours.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** Initial population = 800 bacteria. Doubling time = 3 hours. We need an exponential function: y = a * (b)^x where x is in hours. --- **Step 2: Identify a** At x = 0 hours, population = 800. So a = 800. Thus: y = 800 * (b)^x --- **Step 3: Use doubling time to find b** We know: when x = 3 hours, population = 800 * 2 = 1600. So: 1600 = 800 * (b)^3 --- **Step 4: Solve for b** Divide both sides by 800: 2 = (b)^3 So: b = 2^(1/3) --- **Step 5: Interpret b** b is the growth factor per hour. Since the population doubles every 3 hours, the hourly growth factor is the cube root of 2, which is 2^(1/3). --- **Final answer:** b = 2^(1/3)

8. A linear function models the amount of water in a tank, in gallons, over time in hours. The function has a slope of 7 and passes through the point (0, 19). What is the initial value of the function? Answer: 19 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, 19). At x = 0, the y-coordinate is 19.
   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, 19). Step 3: At x = 0, the y-coordinate is 19. Step 4: Therefore, the initial value is 19. The answer is 19.