Exponential Models

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Answer Key & Explanations

Exponential Models · Grade 11 · Worksheet 1

1. Kaia recorded the number of bacteria in a culture over time: (0, 81), (1, 27), (2, 9), (3, 3). Is an exponential model appropriate? Justify your answer by calculating the ratio between consecutive y-values. Answer: B. yes Solution: Check the x-values: They increase by 1 each time (0, 1, 2, 3), which is constant.
   Full step-by-step solution

   Step 1: Check the x-values: They increase by 1 each time (0, 1, 2, 3), which is constant. Step 2: Calculate the ratios between consecutive y-values: - From (0,81) to (1,27): 27/81 = 1/3 - From (1,27) to (2,9): 9/27 = 1/3 - From (2,9) to (3,3): 3/9 = 1/3 Step 3: Since all ratios equal 1/3, there is a constant multiplicative factor. Step 4: With constant x-differences and constant y-ratios, an exponential model is appropriate. The answer is Yes.

2. Olivia analyzed data: x=1, y=7; x=3, y=63; x=5, y=567; x=7, y=5103. Is an exponential model appropriate? Justify by checking constant ratios. Answer: A. yes Solution: Check if x-values increase by a constant amount: 3-1=2, 5-3=2, 7-5=2. Yes, x increases by 2 each time. Calculate ratios between consecutive y-values: 63÷7=9, 567÷63=9, 5103÷567=9.
   Full step-by-step solution

   Step 1: Check if x-values increase by a constant amount: 3-1=2, 5-3=2, 7-5=2. Yes, x increases by 2 each time. Step 2: Calculate ratios between consecutive y-values: 63÷7=9, 567÷63=9, 5103÷567=9. Step 3: All ratios equal 9, indicating a constant multiplicative factor. Step 4: Since there's a constant ratio when x increases by a constant amount, an exponential model is appropriate. The answer is yes.

3. Emma is monitoring the growth of a rare fungus in a controlled laboratory environment. She records the area covered by the fungus (in square centimeters) at the start of each day for five days: Day 0: 5 cm² Day 1: 15 cm² Day 2: 45 cm² Day 3: 135 cm² Day 4: 405 cm² Determine whether the growth of the fungus is best modeled by a linear function or an exponential function. Justify your answer by explaining the pattern observed in the data. Answer: Exponential model is appropriate because the area increases by a constant ratio of 3 each day, not by a constant difference. Solution: Check for constant difference (linear model). Day 1 - Day 0: 15 - 5 = 10 Day 2 - Day 1: 45 - 15 = 30 Day 3 - Day 2: 135 - 45 = 90 Day 4 - Day 3: 405 - 135 = 270 The differences are 10, 30, 90, 270, which are not constant.
   Full step-by-step solution

   Step 1: Check for constant difference (linear model). Day 1 - Day 0: 15 - 5 = 10 Day 2 - Day 1: 45 - 15 = 30 Day 3 - Day 2: 135 - 45 = 90 Day 4 - Day 3: 405 - 135 = 270 The differences are 10, 30, 90, 270, which are not constant. So linear model is not appropriate. Step 2: Check for constant ratio (exponential model). Day 1 / Day 0: 15 / 5 = 3 Day 2 / Day 1: 45 / 15 = 3 Day 3 / Day 2: 135 / 45 = 3 Day 4 / Day 3: 405 / 135 = 3 The ratio is constant at 3 each day. Step 3: Conclusion. Since the area increases by a constant ratio (multiplying by 3 each day), the growth is best modeled by an exponential function. An exponential model of the form A(t) = 5 * (3)^t would describe this growth, where t is the number of days. The answer is: Exponential model is appropriate because the area increases by a constant ratio of 3 each day, not by a constant difference.

4. Sophia is analyzing the growth of a bacterial culture. She records the number of bacteria at regular intervals of 2 hours and plots the data on a coordinate plane. The table below shows her recorded values: Time (hours): 0, 2, 4, 6, 8 Number of bacteria: 12, 60, 300, 1500, 7500 By examining the graph of this data, determine whether a linear or exponential model is more appropriate. Justify your reasoning using the concept of constant ratio versus constant difference. Answer: Exponential model is appropriate. Solution: Examine the differences between consecutive y-values (number of bacteria). Difference from 0 to 2 hours: 60 - 12 = 48 Difference from 2 to 4 hours: 300 - 60 = 240 Difference from 4 to 6 hours: 1500 - 300 = 1200 Difference from 6 to 8 hours: 7500 - 1500 = 6000 The differences are not constant…
   Full step-by-step solution

   Step 1: Examine the differences between consecutive y-values (number of bacteria). Difference from 0 to 2 hours: 60 - 12 = 48 Difference from 2 to 4 hours: 300 - 60 = 240 Difference from 4 to 6 hours: 1500 - 300 = 1200 Difference from 6 to 8 hours: 7500 - 1500 = 6000 The differences are not constant (48, 240, 1200, 6000), so a linear model is not appropriate. Step 2: Examine the ratios between consecutive y-values. Ratio from 0 to 2 hours: 60 / 12 = 5 Ratio from 2 to 4 hours: 300 / 60 = 5 Ratio from 4 to 6 hours: 1500 / 300 = 5 Ratio from 6 to 8 hours: 7500 / 1500 = 5 The ratio is constant (5) for each 2-hour interval. Step 3: Since the y-values are multiplied by a constant factor of 5 when the x-value increases by a constant amount of 2 hours, the data follows an exponential pattern. The general form is y = a * b^x, where a = 12 (initial value at x=0) and b = 5^(1/2) (since the factor is 5 per 2 hours, the hourly growth factor is sqrt(5)). Conclusion: An exponential model is appropriate because the data has a constant multiplicative ratio (5) rather than a constant additive difference. The answer is exponential model is appropriate.

5. A right triangle is drawn on a coordinate plane with vertices at (0,0), (8,0), and (0,6). A circle is inscribed in this triangle such that it is tangent to all three sides. What is the area of this inscribed circle? (Use π = 3.14) Answer: 12.56 Solution: Identify the triangle's dimensions. The legs are 8 units (horizontal) and 6 units (vertical). Calculate the hypotenuse using the Pythagorean theorem: sqrt(8^2 + 6^2) = sqrt(64 + 36) = sqrt(100) = 10 units.
   Full step-by-step solution

   Step 1: Identify the triangle's dimensions. The legs are 8 units (horizontal) and 6 units (vertical). Step 2: Calculate the hypotenuse using the Pythagorean theorem: sqrt(8^2 + 6^2) = sqrt(64 + 36) = sqrt(100) = 10 units. Step 3: Find the area of the triangle: (1/2) * base * height = (1/2) * 8 * 6 = 24 square units. Step 4: Calculate the semi-perimeter: (8 + 6 + 10)/2 = 24/2 = 12 units. Step 5: Use the formula for inradius (r) of a triangle: r = area / semi-perimeter = 24 / 12 = 2 units. Step 6: Calculate the area of the inscribed circle: π * r^2 = 3.14 * (2)^2 = 3.14 * 4 = 12.56 square units. The answer is 12.56.

6. Dr. Chen is studying the decay of a radioactive isotope used in medical imaging. The isotope has a half-life of 8 hours. If a hospital receives a 120-milligram sample, determine how much of the isotope will remain after 24 hours, using the exponential decay model A(t) = A₀(1/2)^(t/h), where A₀ is the initial amount, t is time in hours, and h is the half-life. Answer: 15 Solution: Initial amount A₀ = 120 mg Half-life h = 8 hours Time elapsed t = 24 hours A(t) = A₀(1/2)^(t/h) A(24) = 120 × (1/2)^(24/8) 24/8 = 3 A(24) = 120 × (1/2)^3 Calculate (1/2)^3 (1/2)^3 = 1/8 A(24) = 120 × 1/8 = 120/8 = 15 The answer is 15 milligrams.
   Full step-by-step solution

   Step 1: Identify the given values Initial amount A₀ = 120 mg Half-life h = 8 hours Time elapsed t = 24 hours Step 2: Use the exponential decay formula A(t) = A₀(1/2)^(t/h) Step 3: Substitute the values A(24) = 120 × (1/2)^(24/8) Step 4: Simplify the exponent 24/8 = 3 A(24) = 120 × (1/2)^3 Step 5: Calculate (1/2)^3 (1/2)^3 = 1/8 Step 6: Multiply by the initial amount A(24) = 120 × 1/8 = 120/8 = 15 The answer is 15 milligrams.

7. 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.