Rate of Change

lessonbunny.com

Answer Key & Explanations

Rate of Change · 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 applying the transformation (x, y) → (2x, 2y) to all vertices of the first triangle. What is the area of the larger triangle? Answer: 96 Solution: The problem says the right triangle has vertices at: A = (0, 0) B = (6, 0) C = (6, 8) The transformation is (x, y) → (2x, 2y).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the vertices of the original triangle** The problem says the right triangle has vertices at: A = (0, 0) B = (6, 0) C = (6, 8) --- **Step 2: Understand the transformation** The transformation is (x, y) → (2x, 2y). This means we multiply both coordinates by 2 (a dilation with scale factor 2 from the origin). --- **Step 3: Apply the transformation to each vertex** A' = (2×0, 2×0) = (0, 0) B' = (2×6, 2×0) = (12, 0) C' = (2×6, 2×8) = (12, 16) So the new triangle has vertices: (0, 0), (12, 0), (12, 16). --- **Step 4: Determine the shape and dimensions of the new triangle** Points (0, 0), (12, 0), (12, 16) form a right triangle with: - Base along x-axis from (0, 0) to (12, 0) → length = 12 - Height along vertical line x = 12 from (12, 0) to (12, 16) → height = 16 --- **Step 5: Calculate the area of the new triangle** Area of a right triangle = (1/2) × base × height Area = (1/2) × 12 × 16 = (1/2) × 192 = 96 --- **Step 6: Alternative check using scale factor effect on area** Original triangle: base = 6, height = 8 Area_original = (1/2) × 6 × 8 = 24 Dilation with scale factor 2 multiplies area by 2² = 4. Area_new = 24 × 4 = 96 Both methods give the same result. --- **Final answer:** 96

2. (2.5 × 10⁶) × (4.0 × 10⁻³) = ? Answer: 10000 Solution: Multiply the coefficients: 2.5 × 4.0 = 10.0 Add the exponents: 6 + (-3) = 3 Combine the results: 10.0 × 10³ Convert to standard form: 10.0 × 1000 = 10000 The answer is 10000.
   Full step-by-step solution

   Step 1: Multiply the coefficients: 2.5 × 4.0 = 10.0 Step 2: Add the exponents: 6 + (-3) = 3 Step 3: Combine the results: 10.0 × 10³ Step 4: Convert to standard form: 10.0 × 1000 = 10000 The answer is 10000.

3. (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.

4. A linear function has a slope of 9 and passes through the point (0, 15). What is the initial value of the function? Answer: 15 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, 15). At x = 0, the y-coordinate is 15.
   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, 15). Step 3: At x = 0, the y-coordinate is 15. Step 4: Therefore, the initial value is 15. The answer is 15.

5. (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.

6. 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

7. Nikau is reading a book for a school project. The book has 230 pages, and Nikau reads 46 pages each day. If Nikau 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: 46 Solution: The rate of change is the number of pages read each day: 46 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: 46 pages per day. The initial value is the number of pages read at the start: 0 pages. So the rate of change is 46 and the initial value is 0.

8. 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.

9. A linear function models the distance Hana jogs in kilometers over time in hours. The function has a slope of 12 and passes through the point (0, 3). What is the initial value of the function? Answer: 3 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, 3). At x = 0, the y-coordinate is 3.
   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, 3). Step 3: At x = 0, the y-coordinate is 3. Step 4: Therefore, the initial value is 3. The answer is 3.