Exponential Models

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Answer Key & Explanations

Exponential Models · Grade 11 · Worksheet 2

1. Isabella recorded the number of bacteria in a culture over time: (0, 27), (1, 81), (2, 243), (3, 729). Determine if an exponential model is appropriate by calculating the ratio between consecutive y-values. Answer: A. yes Solution: Check the ratio between y-values when x increases by 1.
   Full step-by-step solution

   Step 1: Check the ratio between y-values when x increases by 1. Step 2: From x=0 to x=1: 81 ÷ 27 = 3 Step 3: From x=1 to x=2: 243 ÷ 81 = 3 Step 4: From x=2 to x=3: 729 ÷ 243 = 3 Step 5: Since the ratio is constant (3) for each step, an exponential model is appropriate. The answer is yes.

2. Mere recorded the population of bacteria in a lab experiment: Day 0: 12, Day 1: 36, Day 2: 108, Day 3: 324. Is an exponential model appropriate? Justify by calculating the ratio between consecutive y-values. Answer: A. yes Solution: Check the ratio between Day 1 and Day 0: 36 ÷ 12 = 3 Check the ratio between Day 2 and Day 1: 108 ÷ 36 = 3 Check the ratio between Day 3 and Day 2: 324 ÷ 108 = 3 Since the ratio is constant (3) as x increases by 1 each time, an exponential model is appropriate.
   Full step-by-step solution

   Step 1: Check the ratio between Day 1 and Day 0: 36 ÷ 12 = 3 Step 2: Check the ratio between Day 2 and Day 1: 108 ÷ 36 = 3 Step 3: Check the ratio between Day 3 and Day 2: 324 ÷ 108 = 3 Step 4: Since the ratio is constant (3) as x increases by 1 each time, an exponential model is appropriate. The answer is Yes.

3. A biologist is studying a population of bacteria that grows exponentially. The initial population is 500 bacteria, and after 3 hours, the population reaches 4,000 bacteria. Write an exponential function in the form P(t) = P₀e^(kt) that models this growth, where t is time in hours. Answer: P(t) = 500e^((ln(8)/3)t) Solution: - Initial population P₀ = 500 - After t = 3 hours, population P(3) = 4000 - Model: P(t) = P₀ e^(k t) Write the general equation with given initial condition.
   Full step-by-step solution

   Let's go step-by-step. We are given: - Initial population P₀ = 500 - After t = 3 hours, population P(3) = 4000 - Model: P(t) = P₀ e^(k t) --- **Step 1: Write the general equation with given initial condition.** P(t) = 500 e^(k t) --- **Step 2: Use the condition at t = 3 hours to solve for k.** At t = 3, P(3) = 4000: 4000 = 500 e^(k * 3) --- **Step 3: Divide both sides by 500.** 4000 / 500 = e^(3k) 8 = e^(3k) --- **Step 4: Take natural logarithm of both sides.** ln(8) = ln(e^(3k)) ln(8) = 3k --- **Step 5: Solve for k.** k = ln(8) / 3 --- **Step 6: Write the final function.** P(t) = 500 e^( (ln(8)/3) t ) --- **Final answer:** P(t) = 500e^((ln(8)/3)t)

4. Olivia is tracking the growth of a plant species in a controlled experiment. She records the height of the plant (in cm) at the end of each week for five weeks: Week 0: 5 cm Week 1: 10 cm Week 2: 20 cm Week 3: 40 cm Week 4: 80 cm Determine whether a linear model or an exponential model is more appropriate for this data, and justify your reasoning by examining the differences or ratios between consecutive weeks. Answer: Exponential model is appropriate because the ratio of consecutive heights is constant (2). Solution: Calculate the differences between consecutive heights: Week 1 - Week 0: 10 - 5 = 5 Week 2 - Week 1: 20 - 10 = 10 Week 3 - Week 2: 40 - 20 = 20 Week 4 - Week 3: 80 - 40 = 40 The differences are 5, 10, 20, 40, which are not constant.
   Full step-by-step solution

   Step 1: Calculate the differences between consecutive heights: Week 1 - Week 0: 10 - 5 = 5 Week 2 - Week 1: 20 - 10 = 10 Week 3 - Week 2: 40 - 20 = 20 Week 4 - Week 3: 80 - 40 = 40 The differences are 5, 10, 20, 40, which are not constant. So a linear model is not appropriate. Step 2: Calculate the ratios of consecutive heights: Week 1 / Week 0: 10 / 5 = 2 Week 2 / Week 1: 20 / 10 = 2 Week 3 / Week 2: 40 / 20 = 2 Week 4 / Week 3: 80 / 40 = 2 The ratios are all exactly 2, which is constant. Step 3: Since the ratio of consecutive y-values is constant (multiplying by 2 each week), the data follows an exponential growth pattern. Therefore, an exponential model is appropriate. The answer is: Exponential model is appropriate because the ratio of consecutive heights is constant (2).

5. Isabella is a financial analyst tracking the value of a rare collectible. She records the value of the item at the end of each year in the table below. Determine whether the data can be modeled by an exponential function. Justify your reasoning by calculating the appropriate ratios or differences. Year (t) | Value (V) in dollars 0 | 250 1 | 375 2 | 562.50 3 | 843.75 Answer: Yes, exponential model is appropriate because the ratio of successive values is constant (each year the value is multiplied by 1.5). Solution: Check the differences (for linear model): 375 - 250 = 125, 562.50 - 375 = 187.50, 843.75 - 562.50 = 281.25. The differences are not constant, so a linear model is not appropriate.
   Full step-by-step solution

   Step 1: Check the differences (for linear model): 375 - 250 = 125, 562.50 - 375 = 187.50, 843.75 - 562.50 = 281.25. The differences are not constant, so a linear model is not appropriate. Step 2: Check the ratios (for exponential model): 375/250 = 1.5, 562.50/375 = 1.5, 843.75/562.50 = 1.5. The ratio is constant at 1.5. Step 3: Since the value multiplies by the same factor (1.5) each year, the data follows an exponential pattern. The answer is yes, exponential model is appropriate.

6. Mere is monitoring the population of a rare bird species in a protected sanctuary. The table below shows the estimated population recorded at the beginning of each year since 2020 (year 0). Determine whether an exponential model is appropriate for this data, and justify your answer. Year (x): 0, 1, 2, 3, 4 Population (y): 200, 400, 800, 1600, 3200 Answer: Yes, exponential model is appropriate because the ratio of successive y-values is constant (each y is multiplied by 2). Solution: Examine the data: as x increases by 1, y changes from 200 to 400, then 800, then 1600, then 3200. Calculate the ratios of successive y-values: 400/200 = 2, 800/400 = 2, 1600/800 = 2, 3200/1600 = 2.
   Full step-by-step solution

   Step 1: Examine the data: as x increases by 1, y changes from 200 to 400, then 800, then 1600, then 3200. Step 2: Calculate the ratios of successive y-values: 400/200 = 2, 800/400 = 2, 1600/800 = 2, 3200/1600 = 2. Step 3: The ratio is constant (always 2), meaning y is multiplied by the same factor each time x increases by 1. Step 4: A constant multiplicative factor indicates exponential growth, not linear (which would require a constant difference, e.g., adding 200 each time). Step 5: Therefore, an exponential model is appropriate for this data.

7. A biologist is modeling a bacterial culture that triples every 4 hours. If the initial population is 500 bacteria, what will the population be after 12 hours? Round your answer to the nearest whole number. Answer: 13500 Solution: The bacteria triple every 4 hours. Initial population = 500. Determine the number of growth periods in 12 hours Each period = 4 hours.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the growth pattern** The bacteria triple every 4 hours. Initial population = 500. --- **Step 2: Determine the number of growth periods in 12 hours** Each period = 4 hours. Number of periods = 12 / 4 = 3 periods. --- **Step 3: Apply the growth for each period** Growth factor = 3 (triples). After 1 period: 500 × 3 = 1500 After 2 periods: 1500 × 3 = 4500 After 3 periods: 4500 × 3 = 13500 Alternatively, using the exponential formula: Final population = Initial × (growth factor)^(number of periods) Final = 500 × (3)^3 Final = 500 × 27 Final = 13500 --- **Step 4: Rounding** 13500 is already a whole number. --- **Final Answer:** 13500