Graph Proportional

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Answer Key & Explanations

Graph Proportional · Grade 8 · Worksheet 1

1. Noah is filling a swimming pool at a constant rate. The graph of the water volume (in gallons) over time (in minutes) is a straight line passing through the origin and the point (6, 96). What is the slope of this line, and what does it represent in this context? Answer: 16 gallons per minute Solution: The line passes through (0, 0) and (6, 96). For a proportional relationship, slope = y/x for any point (x, y) on the line. Using the point (6, 96): slope = 96/6.
   Full step-by-step solution

   Step 1: The line passes through (0, 0) and (6, 96). For a proportional relationship, slope = y/x for any point (x, y) on the line. Step 2: Using the point (6, 96): slope = 96/6. Step 3: Simplify: 96 ÷ 6 = 16. Step 4: So the slope is 16. Since the slope is the unit rate (change in water volume per minute), it represents that the pool fills at a rate of 16 gallons per minute. The answer is 16 gallons per minute.

2. Mason earns money at a constant rate. After working 7 hours, he earns $42. What is the slope of the line representing his earnings (y) as a function of hours worked (x)? Answer: 6 Solution: The relationship is proportional, so it can be written as y = kx, where k is the slope (unit rate). We know that when x = 7 hours, y = $42.
   Full step-by-step solution

   Step 1: The relationship is proportional, so it can be written as y = kx, where k is the slope (unit rate). Step 2: We know that when x = 7 hours, y = $42. Step 3: Substitute into the equation: 42 = k * 7 Step 4: Solve for k: k = 42 / 7 = 6 Step 5: The slope is 6, meaning Mason earns $6 per hour. The answer is 6.

3. Ava is filling a pool at a constant rate. The proportional relationship between time (t) in minutes and water volume (v) in gallons is given by v = 8t. What is the slope of the graph of this relationship, and what does it represent as the unit rate? Answer: 8 gallons per minute Solution: The equation v = 8t is in the form y = kx, where k is the slope and the constant of proportionality. Identify k: In v = 8t, the coefficient of t is 8, so k = 8. The slope of the graph is 8.
   Full step-by-step solution

   Step 1: The equation v = 8t is in the form y = kx, where k is the slope and the constant of proportionality. Step 2: Identify k: In v = 8t, the coefficient of t is 8, so k = 8. Step 3: The slope of the graph is 8. Step 4: The unit rate is the slope, which represents the change in water volume per minute. Since v is in gallons and t is in minutes, the unit rate is 8 gallons per minute. Final answer: 8 gallons per minute.

4. A fuel tanker truck is being loaded with gasoline at a constant rate. A graph of the proportional relationship shows the amount of gasoline (in gallons) on the y-axis and the time (in minutes) on the x-axis. The line passes through the origin and the point (9, 117). What is the slope of the line, and what does it represent as the unit rate? Answer: 13 Solution: Identify the two points on the line. The line passes through the origin (0,0) and the point (9,117). Recall that slope = (change in y) / (change in x).
   Full step-by-step solution

   Step 1: Identify the two points on the line. The line passes through the origin (0,0) and the point (9,117). Step 2: Recall that slope = (change in y) / (change in x). Step 3: Calculate the change in y: 117 - 0 = 117. Step 4: Calculate the change in x: 9 - 0 = 9. Step 5: Divide the change in y by the change in x: 117 / 9 = 13. Step 6: The slope is 13, which means the unit rate is 13 gallons per minute. Final answer: 13

5. Emma is designing a hiking trail that needs to maintain a consistent slope. The trail rises 180 feet over a horizontal distance of 450 feet. What is the slope of the trail expressed as a simplified fraction? Answer: 2/5 Solution: Identify the vertical change (rise) = 180 feet Identify the horizontal change (run) = 450 feet Calculate slope = rise/run = 180/450 Simplify the fraction by dividing numerator and denominator by their greatest common factor Both 180 and 450 are divisible by 90: 180 ÷ 90 = 2, 450 ÷ 90 = 5 The…
   Full step-by-step solution

   Step 1: Identify the vertical change (rise) = 180 feet Step 2: Identify the horizontal change (run) = 450 feet Step 3: Calculate slope = rise/run = 180/450 Step 4: Simplify the fraction by dividing numerator and denominator by their greatest common factor Step 5: Both 180 and 450 are divisible by 90: 180 ÷ 90 = 2, 450 ÷ 90 = 5 Step 6: The simplified fraction is 2/5 The answer is 2/5.

6. Aroha walks at a constant speed. The distance she walks, y meters, is proportional to the time, x minutes. After 9 minutes, she has walked 45 meters. What is the slope of the graph of y = kx? Answer: 5 Solution: The relationship is y = kx, where k is the slope and unit rate. Use the given point (9, 45). Substitute into y = kx: 45 = k * 9.
   Full step-by-step solution

   Step 1: The relationship is y = kx, where k is the slope and unit rate. Step 2: Use the given point (9, 45). Step 3: Substitute into y = kx: 45 = k * 9. Step 4: Solve for k: k = 45 / 9 = 5. Step 5: The slope is 5, meaning Aroha walks 5 meters per minute. The answer is 5.

7. Liam is designing a wheelchair ramp for his school's new accessibility project. The ramp needs to rise 3 feet over a horizontal distance of 36 feet. What is the slope of the ramp? Express your answer as a simplified fraction. Answer: 1/12 Solution: Slope is defined as the ratio of the vertical change (rise) to the horizontal change (run). So, slope = rise / run. Identify the rise and run from the problem.
   Full step-by-step solution

   Step 1: Understand the slope formula. Slope is defined as the ratio of the vertical change (rise) to the horizontal change (run). So, slope = rise / run. Step 2: Identify the rise and run from the problem. The ramp rises 3 feet, so rise = 3. The horizontal distance is 36 feet, so run = 36. Step 3: Write the slope as a fraction. Slope = 3 / 36. Step 4: Simplify the fraction. Both 3 and 36 can be divided by 3. 3 ÷ 3 = 1 36 ÷ 3 = 12 So, 3/36 simplifies to 1/12. Step 5: Interpret the result. The slope of the ramp is 1/12. This means for every 12 feet of horizontal distance, the ramp rises 1 foot. Final answer: 1/12

8. y = 6x = ? Answer: 6 Solution: The equation is y = 6x. This is in the form y = kx, where k is the constant of proportionality. The slope of a proportional relationship is equal to the constant k.
   Full step-by-step solution

   Step 1: The equation is y = 6x. This is in the form y = kx, where k is the constant of proportionality. Step 2: The slope of a proportional relationship is equal to the constant k. Step 3: Here, k = 6, so the slope is 6. Step 4: As a unit rate, this means for every increase of 1 in x, y increases by 6. The answer is 6.