Graph Proportional

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Graph Proportional · Grade 8 · Worksheet 3

1. A car travels 240 miles in 4 hours. What is the speed in miles per hour? Answer: 60 Solution: To find the speed in miles per hour, we use the formula: speed = distance / time Identify the given values. Distance = 240 miles Time = 4 hours Substitute the values into the formula.
   Full step-by-step solution

   To find the speed in miles per hour, we use the formula: speed = distance / time Step 1: Identify the given values. Distance = 240 miles Time = 4 hours Step 2: Substitute the values into the formula. speed = 240 / 4 Step 3: Perform the division. 240 ÷ 4 = 60 Step 4: State the final answer with units. The speed is 60 miles per hour.

2. A wheelchair ramp is being designed to meet accessibility standards. The ramp forms a right triangle on a coordinate plane with its base along the x-axis from (0,0) to (15,0) and its vertical rise along the y-axis from (0,0) to (0,1.5). What is the slope of the ramp's inclined surface? Answer: 0.1 Solution: Identify the two points that define the ramp's inclined surface: (0,0) and (15,1.5) Recall that slope = (change in y) / (change in x) Calculate the change in y: 1.5 - 0 = 1.5 Calculate the change in x: 15 - 0 = 15 Divide the change in y by the change in x: 1.5 ÷ 15 = 0.1 The slope of the ramp is…
   Full step-by-step solution

   Step 1: Identify the two points that define the ramp's inclined surface: (0,0) and (15,1.5) Step 2: Recall that slope = (change in y) / (change in x) Step 3: Calculate the change in y: 1.5 - 0 = 1.5 Step 4: Calculate the change in x: 15 - 0 = 15 Step 5: Divide the change in y by the change in x: 1.5 ÷ 15 = 0.1 Step 6: The slope of the ramp is 0.1, which means it rises 0.1 units vertically for every 1 unit horizontally.

3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A line is drawn from the origin (0,0) to the point (6,8), forming the hypotenuse. What is the slope of this hypotenuse? Answer: 1.333... or 4/3 Solution: Identify the two points on the hypotenuse. The line goes from the origin (0,0) to the point (6,8). So point 1: (x1, y1) = (0, 0) Point 2: (x2, y2) = (6, 8) Recall the slope formula.
   Full step-by-step solution

   Step 1: Identify the two points on the hypotenuse. The line goes from the origin (0,0) to the point (6,8). So point 1: (x1, y1) = (0, 0) Point 2: (x2, y2) = (6, 8) Step 2: Recall the slope formula. Slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1) Step 3: Substitute the coordinates into the formula. Change in y = 8 - 0 = 8 Change in x = 6 - 0 = 6 Slope = 8 / 6 Step 4: Simplify the fraction. 8/6 = 4/3 Step 5: Convert to decimal if needed. 4 divided by 3 = 1.333... Final Answer: The slope of the hypotenuse is 4/3 or 1.333...

4. Liam is designing a wheelchair ramp for a community center. The ramp needs to rise 3 feet over a horizontal distance of 36 feet. What is the slope of the ramp, expressed as a simplified fraction? Answer: 1/12 Solution: Slope is defined as the vertical rise divided by the horizontal run. Identify the given values. Rise = 3 feet Run = 36 feet Write the slope as a fraction.
   Full step-by-step solution

   Step 1: Understand the slope formula. Slope is defined as the vertical rise divided by the horizontal run. Step 2: Identify the given values. Rise = 3 feet Run = 36 feet Step 3: Write the slope as a fraction. Slope = rise / run = 3 / 36 Step 4: Simplify the fraction. Both 3 and 36 can be divided by 3. 3 ÷ 3 = 1 36 ÷ 3 = 12 Step 5: Write the simplified fraction. 3/36 simplifies to 1/12. Step 6: 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

5. A straight line is drawn through the origin (0,0) on a coordinate plane. The line also passes through the point (9, 63). What is the slope of this line, and what does it represent as the unit rate in a proportional relationship? Answer: 7 Solution: Identify the two points on the line: (0,0) and (9,63). Recall the formula for slope: slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1). Substitute the coordinates: y2 - y1 = 63 - 0 = 63.
   Full step-by-step solution

   Step 1: Identify the two points on the line: (0,0) and (9,63). Step 2: Recall the formula for slope: slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1). Step 3: Substitute the coordinates: y2 - y1 = 63 - 0 = 63. x2 - x1 = 9 - 0 = 9. Step 4: Calculate the slope: 63 / 9 = 7. Step 5: In a proportional relationship y = kx, the slope k is the unit rate. Here, k = 7, so the unit rate is 7. Final Answer: 7

6. Noah is filling jars with homemade lemonade for a school fundraiser. Each jar holds the same amount of lemonade. He notices that after filling 6 jars, he has used 4.2 liters of lemonade. If the relationship between the number of jars and the liters of lemonade used is proportional, what is the constant of proportionality (slope) that represents the liters of lemonade per jar? Express your answer as a decimal. Answer: 0.7 Solution: Identify the two quantities in the proportional relationship. The number of jars is the independent variable (x). The relationship is proportional, so it can be written as y = kx, where k is the constant of proportionality (slope).
   Full step-by-step solution

   Step 1: Identify the two quantities in the proportional relationship. The number of jars is the independent variable (x). The liters of lemonade is the dependent variable (y). Step 2: The relationship is proportional, so it can be written as y = kx, where k is the constant of proportionality (slope). Step 3: Use the given point (6 jars, 4.2 liters) to find k. Substitute x = 6 and y = 4.2 into y = kx: 4.2 = k * 6 Step 4: Solve for k by dividing both sides by 6: k = 4.2 / 6 k = 0.7 Step 5: Interpret the result. The constant of proportionality is 0.7, which means Noah uses 0.7 liters of lemonade per jar. The answer is 0.7.

7. Liam is tracking the growth of his sunflower plant. After 3 weeks, it was 15 inches tall, and after 7 weeks, it was 35 inches tall. If the plant's growth is proportional, what is the slope of the line representing its height over time, and what does this slope represent in the context of the story? Answer: 5 inches per week Solution: Liam's sunflower height is proportional to time, so the relationship is linear. We are given two data points: At week 3, height = 15 inches. At week 7, height = 35 inches.
   Full step-by-step solution

   Step 1: Understand the problem. Liam's sunflower height is proportional to time, so the relationship is linear. We are given two data points: At week 3, height = 15 inches. At week 7, height = 35 inches. Step 2: Recall the slope formula. For a line, slope = (change in height) / (change in time). Change in height = height at week 7 − height at week 3 = 35 − 15 = 20 inches. Change in time = 7 − 3 = 4 weeks. Step 3: Calculate the slope. Slope = 20 inches / 4 weeks = 5 inches/week. Step 4: Interpret the slope in context. The slope represents the rate of growth: the plant grows 5 inches each week. Final answer: 5 inches per week.

8. A line passes through points (3, 7) and (9, 19). Find the slope of this line. Answer: B. 2 Solution: Identify the coordinates: (3, 7) and (9, 19) Use the slope formula: slope = (y2 - y1) ÷ (x2 - x1) Substitute the values: slope = (19 - 7) ÷ (9 - 3) Calculate the numerator: 19 - 7 = 12 Calculate the denominator: 9 - 3 = 6 Divide: 12 ÷ 6 = 2 The correct answer is 2.
   Full step-by-step solution

   Step 1: Identify the coordinates: (3, 7) and (9, 19) Step 2: Use the slope formula: slope = (y2 - y1) ÷ (x2 - x1) Step 3: Substitute the values: slope = (19 - 7) ÷ (9 - 3) Step 4: Calculate the numerator: 19 - 7 = 12 Step 5: Calculate the denominator: 9 - 3 = 6 Step 6: Divide: 12 ÷ 6 = 2 The correct answer is 2.