Exponential Models

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Answer Key & Explanations

Exponential Models · Grade 11 · Worksheet 2

1. Sophia analyzed data: x=0, y=64; x=1, y=32; x=2, y=16; x=3, y=8. Is an exponential model appropriate? Answer: B. yes Solution: Calculate the ratio between consecutive y-values. From x=0 to x=1: 32 ÷ 64 = 0.5 From x=1 to x=2: 16 ÷ 32 = 0.5 From x=2 to x=3: 8 ÷ 16 = 0.5 Since the ratio is constant (0.5) as x increases by 1 each time, an exponential model is appropriate.
   Full step-by-step solution

   Step 1: Calculate the ratio between consecutive y-values. Step 2: From x=0 to x=1: 32 ÷ 64 = 0.5 Step 3: From x=1 to x=2: 16 ÷ 32 = 0.5 Step 4: From x=2 to x=3: 8 ÷ 16 = 0.5 Step 5: Since the ratio is constant (0.5) as x increases by 1 each time, an exponential model is appropriate. The answer is yes.

2. Sophia analyzed data from a chemical reaction experiment. The concentration of substance A was measured every 6 minutes: (0, 81), (6, 27), (12, 9), (18, 3). Is an exponential model appropriate for this data? Justify your answer by calculating ratios. Answer: B. yes Solution: Check if x-values increase by a constant amount. Time intervals: 6-0 = 6, 12-6 = 6, 18-12 = 6 ✓ (constant interval of 6 minutes) 27 ÷ 81 = 1/3 9 ÷ 27 = 1/3 3 ÷ 9 = 1/3 Since all ratios equal 1/3 (a constant), the data shows exponential decay.
   Full step-by-step solution

   Step 1: Check if x-values increase by a constant amount. Time intervals: 6-0 = 6, 12-6 = 6, 18-12 = 6 ✓ (constant interval of 6 minutes) Step 2: Calculate ratios between consecutive y-values: 27 ÷ 81 = 1/3 9 ÷ 27 = 1/3 3 ÷ 9 = 1/3 Step 3: Since all ratios equal 1/3 (a constant), the data shows exponential decay. Step 4: The exponential model is appropriate because there is a constant ratio between y-values when x increases by a constant amount. Conclusion: Yes, an exponential model is appropriate.

3. Sophia is analyzing the growth of a certain type of algae in a controlled pond. She measures the surface area covered by the algae (in square meters) at the beginning of each day for five days. Her data is shown below: Day 0: 12 m² Day 1: 18 m² Day 2: 27 m² Day 3: 40.5 m² Day 4: 60.75 m² Determine whether the data can be modeled by an exponential function. Justify your answer by explaining the pattern in the data, and if appropriate, write an exponential function of the form A(t) = A₀ * r^t that models the area covered after t days. Answer: Yes, exponential; A(t) = 12 * (1.5)^t Solution: Check for constant difference (linear model): Day 1 - Day 0 = 18 - 12 = 6. Day 2 - Day 1 = 27 - 18 = 9. The differences are not constant (6, 9), so it is not linear.
   Full step-by-step solution

   Step 1: Check for constant difference (linear model): Day 1 - Day 0 = 18 - 12 = 6. Day 2 - Day 1 = 27 - 18 = 9. The differences are not constant (6, 9), so it is not linear. Step 2: Check for constant ratio (exponential model): Day 1 / Day 0 = 18/12 = 1.5. Day 2 / Day 1 = 27/18 = 1.5. Day 3 / Day 2 = 40.5/27 = 1.5. Day 4 / Day 3 = 60.75/40.5 = 1.5. The ratio is constant at 1.5, so the data is exponential. Step 3: Write the exponential function. The initial value A₀ at t=0 is 12. The growth factor r is 1.5. Therefore, the function is A(t) = 12 * (1.5)^t. The answer is: Yes, exponential; A(t) = 12 * (1.5)^t.

4. Charlotte is a botanist studying the spread of a fungal infection in a forest. She records the number of infected trees in a specific area each month. The data she collected is shown below: Month (x): 0, 1, 2, 3, 4 Number of infected trees (y): 12, 36, 108, 324, 972 Determine whether a linear model or an exponential model is more appropriate for this data. Justify your reasoning by analyzing the differences or ratios between consecutive terms. Answer: Exponential model is appropriate because the ratio of consecutive y-values is constant (3), not the difference. Solution: Check for a linear pattern by finding the differences between consecutive y-values. Difference from month 0 to 1: 36 - 12 = 24 Difference from month 1 to 2: 108 - 36 = 72 Difference from month 2 to 3: 324 - 108 = 216 Difference from month 3 to 4: 972 - 324 = 648 The differences are 24, 72, 216,…
   Full step-by-step solution

   Step 1: Check for a linear pattern by finding the differences between consecutive y-values. Difference from month 0 to 1: 36 - 12 = 24 Difference from month 1 to 2: 108 - 36 = 72 Difference from month 2 to 3: 324 - 108 = 216 Difference from month 3 to 4: 972 - 324 = 648 The differences are 24, 72, 216, 648, which are not constant. So a linear model is not appropriate. Step 2: Check for an exponential pattern by finding the ratios of consecutive y-values. Ratio from month 0 to 1: 36 / 12 = 3 Ratio from month 1 to 2: 108 / 36 = 3 Ratio from month 2 to 3: 324 / 108 = 3 Ratio from month 3 to 4: 972 / 324 = 3 The ratios are all equal to 3. Since the ratio is constant, an exponential model is appropriate. The answer is: Exponential model is appropriate because the ratio of consecutive y-values is constant (3), not the difference.

5. Isabella collected data on the cooling of a liquid: Time (min) 0, 2, 4, 6; Temperature (°C) 97, 67, 47, 32. Determine if an exponential model is appropriate by calculating the ratios of consecutive temperature values. Answer: A. no Solution: Calculate the ratio between the temperature at 2 minutes and 0 minutes: 67 / 97 ≈ 0.6907 Calculate the ratio between the temperature at 4 minutes and 2 minutes: 47 / 67 ≈ 0.7015 Calculate the ratio between the temperature at 6 minutes and 4 minutes: 32 / 47 ≈ 0.6809 Compare the ratios: 0.6907,…
   Full step-by-step solution

   Step 1: Calculate the ratio between the temperature at 2 minutes and 0 minutes: 67 / 97 ≈ 0.6907 Step 2: Calculate the ratio between the temperature at 4 minutes and 2 minutes: 47 / 67 ≈ 0.7015 Step 3: Calculate the ratio between the temperature at 6 minutes and 4 minutes: 32 / 47 ≈ 0.6809 Step 4: Compare the ratios: 0.6907, 0.7015, and 0.6809 are not constant (they vary by about 0.02) Step 5: Since the ratios are not constant, an exponential model is not appropriate for this data. The answer is no.

6. Emma is monitoring the growth of a sunflower in her garden. She measures its height every 5 days and records the data in the table below. Determine whether the height of the sunflower over time is best modeled by a linear function or an exponential function. Justify your answer based on the pattern in the data. | Days | Height (cm) | |------|-------------| | 0 | 5 | | 5 | 10 | | 10 | 20 | | 15 | 40 | | 20 | 80 | Answer: Exponential model is appropriate because the height multiplies by a constant factor of 2 every 5 days, indicating a constant ratio. Solution: Examine the differences in height between consecutive time points. From day 0 to day 5: 10 - 5 = 5 cm From day 5 to day 10: 20 - 10 = 10 cm From day 10 to day 15: 40 - 20 = 20 cm From day 15 to day 20: 80 - 40 = 40 cm The differences (5, 10, 20, 40) are not constant; they are increasing.
   Full step-by-step solution

   Step 1: Examine the differences in height between consecutive time points. From day 0 to day 5: 10 - 5 = 5 cm From day 5 to day 10: 20 - 10 = 10 cm From day 10 to day 15: 40 - 20 = 20 cm From day 15 to day 20: 80 - 40 = 40 cm The differences (5, 10, 20, 40) are not constant; they are increasing. So a linear model is not appropriate. Step 2: Examine the ratios of consecutive heights. From day 0 to day 5: 10 / 5 = 2 From day 5 to day 10: 20 / 10 = 2 From day 10 to day 15: 40 / 20 = 2 From day 15 to day 20: 80 / 40 = 2 The ratio is constant at 2. This means the height multiplies by a factor of 2 every 5 days. Step 3: Conclusion. Since the data shows a constant multiplicative ratio (2) for equal time intervals, an exponential model is appropriate. The height can be modeled as H(d) = 5 * 2^(d/5), where d is days. The answer is exponential.