Rate of Change

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Answer Key & Explanations

Rate of Change · Grade 8 · Worksheet 3

1. Liam is tracking the growth of his sunflower plant. In the first week, it grew 8 centimeters. Each subsequent week, it grew 75% of the amount it grew the previous week. What is the total height the sunflower will have grown after a very long time, starting from its initial growth? Answer: 32 Solution: In the first week, the plant grew 8 cm. Each week after that, it grows 75% of the previous week's growth.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** In the first week, the plant grew 8 cm. Each week after that, it grows 75% of the previous week's growth. So: - Week 1 growth: 8 cm - Week 2 growth: 8 × 0.75 - Week 3 growth: (8 × 0.75) × 0.75 = 8 × (0.75)^2 - Week 4 growth: 8 × (0.75)^3 - and so on. We want the **total growth** after a very long time. --- **Step 2: Recognize the pattern** This is an **infinite geometric series**: Total growth = 8 + 8 × 0.75 + 8 × (0.75)^2 + 8 × (0.75)^3 + ... --- **Step 3: Recall the infinite geometric series formula** For an infinite geometric series: a + a × r + a × r^2 + ... = a / (1 - r) where |r| < 1. Here: a = 8 (first term) r = 0.75 (common ratio) --- **Step 4: Apply the formula** Total growth = 8 / (1 - 0.75) = 8 / 0.25 = 8 × 4 = 32 --- **Step 5: Interpret the result** After a very long time, the total additional growth from the start approaches **32 cm**. --- **Final Answer:** 32

2. Mason is examining a line graph that shows the total amount of money saved in a bank account over time. The graph is a straight line. The line passes through the point (0, 28) and the point (12, 172). The x-axis represents time in weeks, and the y-axis represents the total savings in dollars. What does the slope of this line represent in the context of the savings account, and what is its value? Answer: The slope represents the amount saved per week, and its value is 12 dollars per week. Solution: Identify the two points on the line: (0, 28) and (12, 172). The point (0, 28) means that at week 0, the account already has $28.
   Full step-by-step solution

   Step 1: Identify the two points on the line: (0, 28) and (12, 172). Step 2: The point (0, 28) means that at week 0, the account already has $28. This is the initial value (y-intercept), which could represent an initial deposit or starting balance. 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 = (172 - 28) / (12 - 0) = 144 / 12 = 12. Step 5: The slope is 12. Since the y-axis is savings (in dollars) and the x-axis is time (in weeks), the slope represents the amount of money saved per week. Step 6: Interpretation: Mason saves $12 each week, starting from an initial balance of $28. The answer is: The slope represents the amount saved per week, and its value is 12 dollars per week.

3. Emma's savings account balance increases by $25 each week. After 10 weeks, the balance is $350. What is the initial value of the function representing the balance over time? Answer: 100 Solution: Let x represent weeks and y represent balance. The rate of change (slope) is 25, so m = 25. The function passes through the point (10, 350).
   Full step-by-step solution

   Step 1: Let x represent weeks and y represent balance. The rate of change (slope) is 25, so m = 25. Step 2: The function passes through the point (10, 350). Step 3: Use the slope-intercept form: y = mx + b. Substitute: 350 = 25(10) + b. Step 4: Simplify: 350 = 250 + b. Step 5: Subtract 250 from both sides: 350 - 250 = b, so b = 100. Step 6: The initial value is the y-intercept, which is 100. The answer is 100.

4. Max is reading a book for a school project. The book has 338 pages, and Max reads 15 pages each day. If Max reads at a constant rate, what is the rate of change (pages per day) and the initial value (pages read before starting, which is 0)? Answer: 15 Solution: The rate of change is the number of pages read each day: 15 pages per day. The initial value is the number of pages read at the start: 0 pages.
   Full step-by-step solution

   The rate of change is the number of pages read each day: 15 pages per day. The initial value is the number of pages read at the start: 0 pages. So the rate of change is 15 and the initial value is 0.

5. A linear function has a slope of 6 and passes through the point (0, 14). What is the initial value of the function? Answer: 14 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, 14). At x = 0, the y-coordinate is 14.
   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, 14). Step 3: At x = 0, the y-coordinate is 14. Step 4: Therefore, the initial value is 14. The answer is 14.

6. Aroha is looking at a graph that shows the total cost of ordering custom T-shirts from a local print shop. The graph is a straight line. The line passes through the point (0, 19) and the point (7, 152). What does the slope of this line represent in the context of ordering T-shirts, and what is its value? Answer: The slope represents the cost per T-shirt, and its value is $19 per T-shirt. Solution: Identify the two points on the line: (0, 19) and (7, 152). The point (0, 19) means that when 0 T-shirts are ordered, the total cost is $19.
   Full step-by-step solution

   Step 1: Identify the two points on the line: (0, 19) and (7, 152). Step 2: The point (0, 19) means that when 0 T-shirts are ordered, the total cost is $19. This is the initial value (y-intercept), which represents a setup fee or fixed cost. 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 = (152 - 19) / (7 - 0) = 133 / 7 = 19. Step 5: The slope is 19. Since the y-axis is total cost (in dollars) and the x-axis is the number of T-shirts, the slope represents the cost per T-shirt. Step 6: Interpretation: Each T-shirt costs $19, in addition to the $19 setup fee. The answer is: The slope represents the cost per T-shirt, and its value is $19 per T-shirt.

7. Isabella is reading a book for a school project. The book has 308 pages, and Isabella reads 29 pages each day. If Isabella reads at a constant rate, what is the rate of change (pages per day) and the initial value (pages read before starting, which is 0)? Answer: 29 Solution: The rate of change is the number of pages read each day: 29 pages per day. The initial value is the number of pages read at the start: 0 pages.
   Full step-by-step solution

   The rate of change is the number of pages read each day: 29 pages per day. The initial value is the number of pages read at the start: 0 pages. So the rate of change is 29 and the initial value is 0.

8. Noah is saving money to buy a new video game. Noah already has $131 saved up. Each week, Noah earns $6 from doing chores and adds it to the savings. Assuming Noah saves at a constant rate, identify the rate of change (weekly savings) and the initial value (starting amount). Answer: 6 Solution: The rate of change is the amount saved each week: $6 per week. The initial value is the amount already saved: $131.
   Full step-by-step solution

   The rate of change is the amount saved each week: $6 per week. The initial value is the amount already saved: $131. So the rate of change is 6 and the initial value is 131.