Functional Relationships

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Answer Key & Explanations

Functional Relationships · Grade 8 · Worksheet 2

1. Emma is comparing two job offers for her summer internship. Company A offers a base pay of $200 per week plus $15 for each project completed. Company B offers a base pay of $150 per week plus $20 for each project completed. Emma wants to know how many projects she would need to complete each week for both companies to pay the same total weekly amount. How many projects would that be? Answer: 10 Solution: Write an expression for Company A's weekly pay: 200 + 15x, where x is the number of projects. Write an expression for Company B's weekly pay: 150 + 20x.
   Full step-by-step solution

   Step 1: Write an expression for Company A's weekly pay: 200 + 15x, where x is the number of projects. Step 2: Write an expression for Company B's weekly pay: 150 + 20x. Step 3: Set the two expressions equal to find when the pay is the same: 200 + 15x = 150 + 20x. Step 4: Subtract 150 from both sides: 50 + 15x = 20x. Step 5: Subtract 15x from both sides: 50 = 5x. Step 6: Divide both sides by 5: x = 10. Step 7: Check: Company A: 200 + 15(10) = 200 + 150 = 350. Company B: 150 + 20(10) = 150 + 200 = 350. The answer is 10 projects.

2. (2.4 × 10⁵) ÷ (6 × 10²) = ? Answer: 400 Solution: Write the expression as (2.4 ÷ 6) × (10⁵ ÷ 10²) Divide the coefficients: 2.4 ÷ 6 = 0.4 Divide the powers of 10: 10⁵ ÷ 10² = 10^(5-2) = 10³ Multiply the results: 0.4 × 10³ = 0.4 × 1000 = 400 The answer is 400.
   Full step-by-step solution

   Step 1: Write the expression as (2.4 ÷ 6) × (10⁵ ÷ 10²) Step 2: Divide the coefficients: 2.4 ÷ 6 = 0.4 Step 3: Divide the powers of 10: 10⁵ ÷ 10² = 10^(5-2) = 10³ Step 4: Multiply the results: 0.4 × 10³ = 0.4 × 1000 = 400 The answer is 400.

3. A rectangular prism has a length of 12 cm, width of 8 cm, and height of 5 cm. If you double all three dimensions to create a larger prism, how many times greater will the volume of the new prism be compared to the original? Answer: 8 Solution: Find the volume of the original prism. The formula for the volume of a rectangular prism is: Volume = length × width × height Original length = 12 cm, width = 8 cm, height = 5 cm So, original volume = 12 × 8 × 5 First, 12 × 8 = 96 Then, 96 × 5 = 480 Original volume = 480 cubic cm.
   Full step-by-step solution

   Step 1: Find the volume of the original prism. The formula for the volume of a rectangular prism is: Volume = length × width × height Original length = 12 cm, width = 8 cm, height = 5 cm So, original volume = 12 × 8 × 5 First, 12 × 8 = 96 Then, 96 × 5 = 480 Original volume = 480 cubic cm. Step 2: Find the dimensions of the new prism after doubling. New length = 12 × 2 = 24 cm New width = 8 × 2 = 16 cm New height = 5 × 2 = 10 cm Step 3: Find the volume of the new prism. New volume = 24 × 16 × 10 First, 24 × 16 = 384 Then, 384 × 10 = 3840 New volume = 3840 cubic cm. Step 4: Compare the new volume to the original volume. Volume ratio = New volume ÷ Original volume Volume ratio = 3840 ÷ 480 3840 ÷ 480 = 8 Step 5: Explain the reasoning. When all three dimensions are doubled, each dimension is multiplied by 2, so the volume is multiplied by 2 × 2 × 2 = 8. This matches our calculation: the new volume is 8 times the original volume. Final answer: 8

4. Isabella is training for a charity run. She records her distance (in miles) over time (in minutes) during a long training session. For the first 15 minutes, she runs at a steady pace, covering 0.1 miles each minute. Then, she slows down and covers only 0.05 miles each minute for the next 20 minutes. Finally, she walks the last 10 minutes, covering a constant 0.02 miles each minute. Describe how the total distance she has covered changes over time during each of the three intervals. Is the relationship increasing, decreasing, or constant in each interval? Answer: During the first 15 minutes, the total distance is increasing at a constant rate. During the next 20 minutes, the total distance is still increasing, but at a slower constant rate. During the last 10 minutes, the total distance continues to increase, but at an even slower constant rate. Solution: Isabella is tracking total distance over time. As time passes, she keeps moving forward, so her total distance always increases (never decreases or stays flat). First interval (0 to 15 minutes): She runs at 0.1 miles per minute.
   Full step-by-step solution

   Step 1: Understand the context. Isabella is tracking total distance over time. As time passes, she keeps moving forward, so her total distance always increases (never decreases or stays flat). Step 2: First interval (0 to 15 minutes): She runs at 0.1 miles per minute. This is a constant positive rate, so total distance increases steadily (linear increase). The relationship is increasing and constant rate. Step 3: Second interval (15 to 35 minutes): She runs at 0.05 miles per minute, which is still positive but slower. Total distance continues to increase, but the rate of increase is slower. The relationship is still increasing, but at a slower constant rate. Step 4: Third interval (35 to 45 minutes): She walks at 0.02 miles per minute, still positive but even slower. Total distance keeps increasing, but at the slowest constant rate so far. Conclusion: In all three intervals, the total distance is increasing (never decreasing or constant), but the rate of increase changes from faster to slower. The answer describes each interval's behavior as increasing with a constant rate (different for each).

5. Emma is tracking the height of a plant over 8 weeks. The height in centimeters is given by the function h(w) = 3w + 5, where w is the number of weeks since planting. Describe how the height changes as w increases from 0 to 8. Answer: The height increases at a constant rate of 3 cm per week. Solution: Identify the function: h(w) = 3w + 5. This is a linear function in slope-intercept form (y = mx + b), where m = 3 and b = 5.
   Full step-by-step solution

   Step 1: Identify the function: h(w) = 3w + 5. This is a linear function in slope-intercept form (y = mx + b), where m = 3 and b = 5. Step 2: The slope m = 3 means that for every increase of 1 in w (each week), the height h increases by 3 centimeters. Step 3: The y-intercept b = 5 means the initial height at week 0 is 5 cm. Step 4: As w increases from 0 to 8, the height increases steadily. For example, at w = 0, h = 5 cm; at w = 4, h = 3(4) + 5 = 17 cm; at w = 8, h = 3(8) + 5 = 29 cm. Step 5: Since the slope is positive and constant, the function is increasing at a constant rate. The height increases by 3 cm each week. The answer is: The height increases at a constant rate of 3 cm per week.

6. A rocket travels at a constant speed of 2.5 × 10^4 kilometers per hour. How many hours will it take for the rocket to travel a distance of 3.75 × 10^5 kilometers? Answer: 15 Solution: Write the formula for time: time = distance ÷ speed Substitute the given values: time = (3.75 × 10^5) ÷ (2.5 × 10^4) Divide the coefficients: 3.75 ÷ 2.5 = 1.5 Divide the powers of 10: 10^5 ÷ 10^4 = 10^(5-4) = 10^1 Combine the results: 1.5 × 10^1 Convert to standard form: 1.5 × 10 = 15 The answer…
   Full step-by-step solution

   Step 1: Write the formula for time: time = distance ÷ speed Step 2: Substitute the given values: time = (3.75 × 10^5) ÷ (2.5 × 10^4) Step 3: Divide the coefficients: 3.75 ÷ 2.5 = 1.5 Step 4: Divide the powers of 10: 10^5 ÷ 10^4 = 10^(5-4) = 10^1 Step 5: Combine the results: 1.5 × 10^1 Step 6: Convert to standard form: 1.5 × 10 = 15 The answer is 15.