Graph Proportional

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Answer Key & Explanations

Graph Proportional · Grade 8 · Worksheet 3

1. Tane is filling bags of rice for a fundraiser. The amount of rice he packs is proportional to the time he works. After 9 minutes, he has packed 27 kilograms of rice. What is the slope of the line representing kilograms of rice packed per minute, and what does it represent in this situation? Answer: 3 Solution: Identify the quantities. Time (x) is 9 minutes, rice packed (y) is 27 kilograms. Since the relationship is proportional, the equation is y = kx, where k is the slope (unit rate).
   Full step-by-step solution

   Step 1: Identify the quantities. Time (x) is 9 minutes, rice packed (y) is 27 kilograms. Step 2: Since the relationship is proportional, the equation is y = kx, where k is the slope (unit rate). Step 3: Substitute the known values: 27 = k * 9. Step 4: Solve for k by dividing both sides by 9: k = 27 / 9 = 3. Step 5: The slope is 3, which means Tane packs 3 kilograms of rice per minute. The answer is 3.

2. A research satellite is traveling through space at a constant speed. The distance it travels (in kilometers) is proportional to the time it travels (in hours). After 3.5 hours, the satellite has traveled 2.8 × 10⁵ kilometers. How many kilometers will the satellite travel in 8 hours? Express your answer in scientific notation. Answer: 6.4e5 Solution: Find the constant of proportionality (speed). Since distance = speed × time, speed = distance ÷ time.
   Full step-by-step solution

   Step 1: Find the constant of proportionality (speed). Since distance = speed × time, speed = distance ÷ time. Step 2: Speed = (2.8 × 10⁵ km) ÷ (3.5 hours) = 0.8 × 10⁵ km/hour = 8 × 10⁴ km/hour Step 3: Now calculate the distance for 8 hours: distance = speed × time = (8 × 10⁴ km/hour) × (8 hours) Step 4: Distance = 64 × 10⁴ km = 6.4 × 10⁵ km The answer is 6.4 × 10⁵ kilometers.

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. A graph shows a straight line passing through the origin and the point (9, 63). The line represents the proportional relationship between the number of hours Tane spends reading and the total number of pages he reads. What is the slope of this line, and what does it represent as the unit rate? Answer: 7 Solution: Identify the two points on the line. The line passes through the origin (0,0) and the point (9,63). Step 2: Use the slope formula: slope = (change in y) / (change in x) = (63 - 0) / (9 - 0) = 63/9.
   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 (9,63). Step 2: Use the slope formula: slope = (change in y) / (change in x) = (63 - 0) / (9 - 0) = 63/9. Step 3: Simplify the fraction: 63 divided by 9 equals 7. Step 4: Interpret the slope as the unit rate: For every 1 hour of reading, Tane reads 7 pages. Final Answer: 7

5. Liam is tracking his savings for a new video game. He starts with $15 and saves $8 each week. After 4 weeks, he graphs his total savings over time and notices the line passes through points (2, 31) and (5, 55). What is the slope of Liam's savings graph, and what does it represent in this situation? Answer: 8, which represents the amount he saves each week Solution: Liam starts with $15 and saves $8 each week. We are told that after 4 weeks, he graphs his total savings over time, and the line passes through (2, 31) and (5, 55).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** Liam starts with $15 and saves $8 each week. We are told that after 4 weeks, he graphs his total savings over time, and the line passes through (2, 31) and (5, 55). We need to find the slope of the graph and interpret it. --- **Step 2: Recall the slope formula** The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \((x_1, y_1) = (2, 31)\) and \((x_2, y_2) = (5, 55)\). --- **Step 3: Calculate the slope** \[ m = \frac{55 - 31}{5 - 2} \] \[ m = \frac{24}{3} \] \[ m = 8 \] So the slope is 8. --- **Step 4: Interpret the slope** The graph plots **weeks** (x-axis) vs **total savings** (y-axis). Slope = change in savings / change in time (weeks). So slope = 8 dollars per week. This matches the given information: Liam saves $8 each week. --- **Step 5: Verify with the initial condition** Initial savings = $15 at week 0. If slope = 8, then equation is: Total savings = 15 + 8 × (number of weeks). Check point (2, 31): At week 2, savings = 15 + 8×2 = 15 + 16 = 31. ✓ Check point (5, 55): At week 5, savings = 15 + 8×5 = 15 + 40 = 55. ✓ --- **Final Answer:** The slope is 8, which represents the amount he saves each week.

6. A line passes through points (2, 5) and (6, 17). Find the slope of this line. Answer: 3 Solution: To find the slope of a line that passes through two points, we use the slope formula: slope = (y2 - y1) / (x2 - x1) Identify the coordinates of the two points.
   Full step-by-step solution

   To find the slope of a line that passes through two points, we use the slope formula: slope = (y2 - y1) / (x2 - x1) Step 1: Identify the coordinates of the two points. Point 1: (x1, y1) = (2, 5) Point 2: (x2, y2) = (6, 17) Step 2: Substitute the coordinates into the slope formula. slope = (17 - 5) / (6 - 2) Step 3: Perform the subtraction in the numerator and the denominator. Numerator: 17 - 5 = 12 Denominator: 6 - 2 = 4 Step 4: Divide the numerator by the denominator. slope = 12 / 4 Step 5: Simplify the fraction. 12 divided by 4 equals 3. Therefore, the slope of the line is 3.

7. 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

8. 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.