Understand Slope

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Understand Slope · Grade 8 · Worksheet 3

1. Liam is designing a wheelchair ramp for a community center. 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 platform that is 15 inches high, what is the minimum horizontal length the ramp must have? Answer: 180 Solution: The code says: for every 3 inches of vertical rise, the ramp must extend at least 36 inches horizontally.
   Full step-by-step solution

   Let's solve this step by step. --- **Step 1: Understand the ratio from the building code** 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 inches vertical → 36 inches horizontal So the ratio (vertical / horizontal) is: 3 / 36 --- **Step 2: Simplify the ratio** 3 / 36 = 1 / 12 This means for 1 inch of vertical rise, the horizontal run must be at least 12 inches. --- **Step 3: Apply the ratio to the given vertical rise** The platform height is 15 inches (vertical rise). Using the ratio: Horizontal length = vertical rise × (horizontal / vertical) From the ratio: horizontal / vertical = 36 / 3 = 12 So: Horizontal length = 15 × 12 --- **Step 4: Calculate** 15 × 12 = 180 inches --- **Step 5: Conclusion** The minimum horizontal length the ramp must have is 180 inches. --- **Final Answer:** 180

2. A line is drawn on a coordinate plane that passes through points (3, 7) and (9, 19). Another line is drawn perpendicular to this line and passes through point (6, 13). What is the slope of this perpendicular line? Answer: -1/2 Solution: Find the slope of the first line using the formula (y2 - y1)/(x2 - x1) Using points (3, 7) and (9, 19): slope = (19 - 7)/(9 - 3) = 12/6 = 2 For perpendicular lines, the slopes are negative reciprocals of each other The negative reciprocal of 2 is -1/2 Therefore, the slope of the perpendicular…
   Full step-by-step solution

   Step 1: Find the slope of the first line using the formula (y2 - y1)/(x2 - x1) Step 2: Using points (3, 7) and (9, 19): slope = (19 - 7)/(9 - 3) = 12/6 = 2 Step 3: For perpendicular lines, the slopes are negative reciprocals of each other Step 4: The negative reciprocal of 2 is -1/2 Step 5: Therefore, the slope of the perpendicular line is -1/2

3. Liam is designing a wheelchair ramp for a community center. The building code requires that for every 12 inches of horizontal distance, the ramp can only rise 1 inch. If the ramp needs to reach a doorway that is 15 inches above ground level, what is the minimum horizontal distance in feet that Liam's ramp must cover? Answer: 15 Solution: The building code says: for every 12 inches of horizontal distance, the ramp can rise 1 inch. Rise / Run = 1 inch / 12 inches. The ramp must reach a doorway 15 inches above ground level.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the slope requirement** The building code says: for every 12 inches of horizontal distance, the ramp can rise 1 inch. This means the ratio of rise to run is: Rise / Run = 1 inch / 12 inches. --- **Step 2: Identify the given rise** The ramp must reach a doorway 15 inches above ground level. So, Rise = 15 inches. --- **Step 3: Set up the proportion** Let \( R \) = horizontal distance in inches. From the ratio: 1 inch rise / 12 inches horizontal = 15 inches rise / \( R \) inches horizontal. So: 1 / 12 = 15 / \( R \) --- **Step 4: Solve for \( R \)** Cross-multiply: 1 × \( R \) = 12 × 15 \( R \) = 180 inches. --- **Step 5: Convert inches to feet** Since 1 foot = 12 inches, \( R \) in feet = 180 / 12 = 15 feet. --- **Step 6: Conclusion** The minimum horizontal distance the ramp must cover is 15 feet. --- **Final answer:** 15

4. (-2x + 8) ÷ 2 = ? Answer: -x + 4 Solution: Start with the expression (-2x + 8) ÷ 2 Apply division to each term: (-2x) ÷ 2 + 8 ÷ 2 Simplify each term: -2x ÷ 2 = -x and 8 ÷ 2 = 4 Combine the simplified terms: -x + 4 The answer is -x + 4.
   Full step-by-step solution

   Step 1: Start with the expression (-2x + 8) ÷ 2 Step 2: Apply division to each term: (-2x) ÷ 2 + 8 ÷ 2 Step 3: Simplify each term: -2x ÷ 2 = -x and 8 ÷ 2 = 4 Step 4: Combine the simplified terms: -x + 4 The answer is -x + 4.

5. Find the slope of the line passing through points (3, 9) and (7, 25). Answer: 4 Solution: Identify the coordinates: (3, 9) and (7, 25). Use the slope formula: slope = (y₂ - y₁) / (x₂ - x₁). Substitute the values: slope = (25 - 9) / (7 - 3).
   Full step-by-step solution

   Step 1: Identify the coordinates: (3, 9) and (7, 25). Step 2: Use the slope formula: slope = (y₂ - y₁) / (x₂ - x₁). Step 3: Substitute the values: slope = (25 - 9) / (7 - 3). Step 4: Simplify the numerator: 25 - 9 = 16. Step 5: Simplify the denominator: 7 - 3 = 4. Step 6: Divide: 16 ÷ 4 = 4. The slope of the line is 4.

6. Maria is hiking up a mountain trail. The trail rises 400 feet in elevation over a horizontal distance of 1,200 feet. After a rest stop, the trail continues with a steeper section that rises 350 feet over a horizontal distance of 700 feet. What is the slope of the steeper section of the trail as a simplified fraction? Answer: 1/2 Solution: Identify the vertical change (rise) = 350 feet Identify the horizontal change (run) = 700 feet Write the slope as rise/run = 350/700 Simplify the fraction by dividing numerator and denominator by 50: 350 ÷ 50 = 7, 700 ÷ 50 = 14 Simplify further by dividing numerator and denominator by 7: 7 ÷ 7 =…
   Full step-by-step solution

   Step 1: Identify the vertical change (rise) = 350 feet Step 2: Identify the horizontal change (run) = 700 feet Step 3: Write the slope as rise/run = 350/700 Step 4: Simplify the fraction by dividing numerator and denominator by 50: 350 ÷ 50 = 7, 700 ÷ 50 = 14 Step 5: Simplify further by dividing numerator and denominator by 7: 7 ÷ 7 = 1, 14 ÷ 7 = 2 Step 6: The simplified slope is 1/2 The answer is 1/2.

7. (-4 - 2)² ÷ (3 × -2) = ? Answer: -6 Solution: Calculate inside the parentheses: (-4 - 2) = -6 Apply the exponent: (-6)² = 36 Calculate the denominator: (3 × -2) = -6 Divide: 36 ÷ (-6) = -6 The answer is -6.
   Full step-by-step solution

   Step 1: Calculate inside the parentheses: (-4 - 2) = -6 Step 2: Apply the exponent: (-6)² = 36 Step 3: Calculate the denominator: (3 × -2) = -6 Step 4: Divide: 36 ÷ (-6) = -6 The answer is -6.

8. (2x - 8)/4 = 3 Answer: 10 Solution: Multiply both sides by 4 to eliminate the fraction: (2x - 8)/4 × 4 = 3 × 4 This simplifies to: 2x - 8 = 12 Add 8 to both sides: 2x - 8 + 8 = 12 + 8 This simplifies to: 2x = 20 Divide both sides by 2: 2x/2 = 20/2 This gives: x = 10 The answer is 10.
   Full step-by-step solution

   Step 1: Multiply both sides by 4 to eliminate the fraction: (2x - 8)/4 × 4 = 3 × 4 Step 2: This simplifies to: 2x - 8 = 12 Step 3: Add 8 to both sides: 2x - 8 + 8 = 12 + 8 Step 4: This simplifies to: 2x = 20 Step 5: Divide both sides by 2: 2x/2 = 20/2 Step 6: This gives: x = 10 The answer is 10.

9. (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.