Understand Slope

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Understand Slope · Grade 8 · Worksheet 1

1. Liam is designing a wheelchair ramp for his school's new accessibility project. The building code requires that for every 3 inches of vertical rise, the ramp must extend at least 36 inches horizontally. If the ramp needs to reach a doorway that is 15 inches above ground level, what is the minimum horizontal length the ramp must extend? Answer: 180 inches Solution: 1. This means the ratio of vertical rise to horizontal run is 3 : 36. 2.
   Full step-by-step solution

   Let's go step by step. 1. Understand the requirement: The code says: for every 3 inches of vertical rise, the ramp must extend at least 36 inches horizontally. This means the ratio of vertical rise to horizontal run is 3 : 36. 2. Write the ratio as a fraction: Vertical rise / Horizontal run = 3 / 36. This simplifies to 1 / 12 (since 3 ÷ 3 = 1 and 36 ÷ 3 = 12). So for every 1 inch of rise, you need 12 inches of horizontal run. 3. Apply this to the given rise: The doorway is 15 inches above ground level, so the vertical rise = 15 inches. Let the horizontal length be L inches. 4. Set up the proportion: Using the ratio: rise / run = 1 / 12 15 / L = 1 / 12 5. Solve for L: Cross-multiply: 15 × 12 = 1 × L L = 180 6. Conclusion: The minimum horizontal length the ramp must extend is 180 inches. Answer: 180 inches

2. (2x + 8) ÷ 2 = 10 Answer: 6 Solution: The equation is (2x + 8) ÷ 2 = 10 Multiply both sides by 2 to eliminate the division: 2x + 8 = 20 Subtract 8 from both sides to isolate the term with x: 2x = 12 Divide both sides by 2 to solve for x: x = 6 The answer is 6.
   Full step-by-step solution

   Step 1: The equation is (2x + 8) ÷ 2 = 10 Step 2: Multiply both sides by 2 to eliminate the division: 2x + 8 = 20 Step 3: Subtract 8 from both sides to isolate the term with x: 2x = 12 Step 4: Divide both sides by 2 to solve for x: x = 6 The answer is 6.

3. A rectangular garden is drawn on a coordinate plane with corners at (1, 2), (7, 2), (7, 8), and (1, 8). A diagonal path is drawn from the bottom-left corner to the top-right corner. What is the slope of this diagonal path? Answer: 1 Solution: Identify the two points for the diagonal path: bottom-left corner (1, 2) and top-right corner (7, 8).
   Full step-by-step solution

   Step 1: Identify the two points for the diagonal path: bottom-left corner (1, 2) and top-right corner (7, 8). Step 2: Calculate the change in y (vertical change): 8 - 2 = 6 Step 3: Calculate the change in x (horizontal change): 7 - 1 = 6 Step 4: Calculate the slope using the formula: slope = (change in y)/(change in x) = 6/6 Step 5: Simplify the fraction: 6/6 = 1 The answer is 1.

4. (3x + 7) = 22 Answer: 5 Solution: We are solving the equation: (3x + 7) = 22. The equation is 3x + 7 = 22. We want to isolate the term with x.
   Full step-by-step solution

   We are solving the equation: (3x + 7) = 22. Step 1: The equation is 3x + 7 = 22. We want to isolate the term with x. First, subtract 7 from both sides to remove the constant term on the left. 3x + 7 - 7 = 22 - 7 This simplifies to: 3x = 15. Step 2: Now, we have 3x = 15. To solve for x, divide both sides by 3. 3x / 3 = 15 / 3 This gives: x = 5. Step 3: Check the solution by substituting x = 5 back into the original equation: 3(5) + 7 = 15 + 7 = 22, which matches the right-hand side of the original equation. Therefore, the correct answer is x = 5.

5. Liam is designing a wheelchair ramp for a community center. The building code requires that the ramp must rise no more than 1 foot for every 12 feet of horizontal distance. If the ramp needs to reach a platform that is 3 feet high, what is the minimum horizontal distance, in feet, that the ramp must cover? Answer: 36 Solution: Rise ≤ 1 foot for every 12 feet of horizontal distance. This means the slope (rise / run) ≤ 1/12.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the building code requirement** The code says: Rise ≤ 1 foot for every 12 feet of horizontal distance. This means the slope (rise / run) ≤ 1/12. --- **Step 2: Write the slope inequality** Let: - Rise = 3 feet (height of platform) - Run = x feet (horizontal distance we need to find) Slope = Rise / Run = 3 / x The requirement: 3 / x ≤ 1 / 12 --- **Step 3: Solve the inequality** Multiply both sides by x (x > 0, so inequality direction stays the same): 3 ≤ (1/12) * x Multiply both sides by 12: 3 * 12 ≤ x 36 ≤ x So: x ≥ 36 --- **Step 4: Interpret the result** The minimum horizontal distance occurs when x = 36 feet. This is because if x is smaller than 36, the slope would be steeper than 1/12, violating the code. --- **Final Answer:** 36

6. Mason is analyzing the steepness of a ramp on a coordinate grid. The ramp starts at point (12, 8) and ends at point (20, 32). What is the slope of the ramp? Answer: 3 Solution: Identify the two points on the line: (12, 8) and (20, 32). Use the slope formula: slope = (y2 - y1) / (x2 - x1). Substitute the values: y2 = 32, y1 = 8, x2 = 20, x1 = 12.
   Full step-by-step solution

   Step 1: Identify the two points on the line: (12, 8) and (20, 32). Step 2: Use the slope formula: slope = (y2 - y1) / (x2 - x1). Step 3: Substitute the values: y2 = 32, y1 = 8, x2 = 20, x1 = 12. Step 4: Calculate the rise: 32 - 8 = 24. Step 5: Calculate the run: 20 - 12 = 8. Step 6: Divide rise by run: 24 / 8 = 3. The slope of the ramp is 3.

7. A line passes through the points (2, 8) and (6, 24). What is the slope of this line? Answer: 4 Solution: Identify the coordinates of the two points: (2, 8) and (6, 24) Use the slope formula: slope = (y2 - y1) / (x2 - x1) Substitute the values: slope = (24 - 8) / (6 - 2) Calculate the numerator: 24 - 8 = 16 Calculate the denominator: 6 - 2 = 4 Divide: 16 ÷ 4 = 4 The answer is 4.
   Full step-by-step solution

   Step 1: Identify the coordinates of the two points: (2, 8) and (6, 24) Step 2: Use the slope formula: slope = (y2 - y1) / (x2 - x1) Step 3: Substitute the values: slope = (24 - 8) / (6 - 2) Step 4: Calculate the numerator: 24 - 8 = 16 Step 5: Calculate the denominator: 6 - 2 = 4 Step 6: Divide: 16 ÷ 4 = 4 The answer is 4.

8. Oliver is hiking up a mountain trail. The trail rises 24 meters vertically for every 4 meters horizontally. What is the slope of the trail? Answer: 6 Solution: Identify the rise and run. The vertical rise is 24 meters, and the horizontal run is 4 meters. Use the slope formula: slope = rise / run.
   Full step-by-step solution

   Step 1: Identify the rise and run. The vertical rise is 24 meters, and the horizontal run is 4 meters. Step 2: Use the slope formula: slope = rise / run. Step 3: Substitute the values: slope = 24 / 4. Step 4: Simplify: 24 / 4 = 6. The slope of the trail is 6.

9. Find the slope of the line passing through points (3, 7) and (5, 13) = ? Answer: 3 Solution: Identify the coordinates: (3, 7) and (5, 13) Use the slope formula: slope = (y₂ - y₁) ÷ (x₂ - x₁) Substitute the values: slope = (13 - 7) ÷ (5 - 3) Calculate the differences: slope = 6 ÷ 2 Simplify: slope = 3 The answer is 3.
   Full step-by-step solution

   Step 1: Identify the coordinates: (3, 7) and (5, 13) Step 2: Use the slope formula: slope = (y₂ - y₁) ÷ (x₂ - x₁) Step 3: Substitute the values: slope = (13 - 7) ÷ (5 - 3) Step 4: Calculate the differences: slope = 6 ÷ 2 Step 5: Simplify: slope = 3 The answer is 3.