Regression Exponential

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Answer Key & Explanations

Regression Exponential · Grade 11 · Worksheet 3

1. A scatter plot shows the cooling of a liquid over time. The temperature data points are: (0, 95), (2, 76), (4, 60.8), and (6, 48.64). When plotted on a semi-log graph with temperature on the logarithmic y-axis and time on the linear x-axis, the points form a straight line. Determine the exponential regression model in the form T(t) = a * b^t that fits this cooling data, where T is temperature in degrees Celsius and t is time in minutes. Answer: 95*0.8^t Solution: Step 1: Identify the pattern in the data: (0, 95), (2, 76), (4, 60.8), (6, 48.64) Step 2: Calculate the ratio between consecutive temperatures: 76/95 = 0.8, 60.8/76 = 0.8, 48.64/60.8 = 0.8 Step 3: Since the ratio is constant at 0.8 over 2-minute intervals, the hourly decay factor is 0.8 Step 4:…
   Full step-by-step solution

   Step 1: Identify the pattern in the data: (0, 95), (2, 76), (4, 60.8), (6, 48.64) Step 2: Calculate the ratio between consecutive temperatures: 76/95 = 0.8, 60.8/76 = 0.8, 48.64/60.8 = 0.8 Step 3: Since the ratio is constant at 0.8 over 2-minute intervals, the hourly decay factor is 0.8 Step 4: For exponential models, when time increases by 1 unit, the multiplier is the square root of 0.8 because 2 minutes is our time unit: sqrt(0.8) = 0.8^(1/2) Step 5: The initial value at t=0 is 95, so a = 95 Step 6: The base b = 0.8^(1/2) = 0.8^0.5 ≈ 0.8944, but we can also write the model using the 2-minute decay directly: T(t) = 95 * (0.8)^(t/2) Step 7: Alternatively, we can verify with the data points: At t=2: 95 * 0.8^(2/2) = 95 * 0.8 = 76 ✓ Step 8: The exponential regression model is T(t) = 95 * 0.8^(t/2) or equivalently T(t) = 95 * (0.8^(1/2))^t Step 9: Since the problem asks for form a * b^t, we use b = 0.8^(1/2) ≈ 0.8944, giving T(t) = 95 * 0.8944^t Step 10: The exact form is T(t) = 95 * 0.8^(t/2)

2. A marine biologist is studying the population growth of an invasive species of jellyfish in a coastal ecosystem. After collecting data for 6 months, she records the following population counts: Month 1: 120, Month 2: 180, Month 3: 270, Month 4: 405, Month 5: 608, Month 6: 912. Determine the exponential regression model of the form y = ab^x that best fits this data, where x is the month number and y is the population. Round the coefficients a and b to three decimal places. Answer: y = 80.000 * 1.500^x Solution: Exponential regression is used to model situations where a quantity grows or decays at a rate proportional to its current value.
   Full step-by-step solution

   Exponential regression is used to model situations where a quantity grows or decays at a rate proportional to its current value. This creates a characteristic pattern where the ratio between successive terms remains relatively constant. In real-world applications like population studies, this model helps predict future growth and understand the dynamics of the system. The regression process finds the best-fitting curve by minimizing the squared differences between observed data and the model's predictions.

3. A marine biologist is studying the growth of a new coral species in a reef ecosystem. After collecting data for 6 months, she records the coral colony size: Month 1: 12 cm², Month 2: 18 cm², Month 3: 27 cm², Month 4: 40.5 cm², Month 5: 60.75 cm², Month 6: 91.125 cm². Determine the exponential regression model that best fits this growth pattern, expressing it in the form y = ab^x, where y is the coral area in cm² and x is the time in months. Answer: y = 8(1.5)^x Solution: 1 | 12 2 | 18 3 | 27 4 | 40.5 5 | 60.75 6 | 91.125 In exponential growth, y = a * b^x, the ratio between consecutive y-values should be constant (equal to b).
   Full step-by-step solution

   Let's find the exponential regression model in the form y = a * b^x from the given data. --- **Step 1: Understand the data** We have: Month (x) | Area y (cm²) 1 | 12 2 | 18 3 | 27 4 | 40.5 5 | 60.75 6 | 91.125 --- **Step 2: Check for constant growth factor** In exponential growth, y = a * b^x, the ratio between consecutive y-values should be constant (equal to b). Let's compute ratios: From x=1 to x=2: 18 / 12 = 1.5 From x=2 to x=3: 27 / 18 = 1.5 From x=3 to x=4: 40.5 / 27 = 1.5 From x=4 to x=5: 60.75 / 40.5 = 1.5 From x=5 to x=6: 91.125 / 60.75 = 1.5 Yes, the growth factor b = 1.5. --- **Step 3: Find the initial value a** We know y = a * (1.5)^x. Use one data point to solve for a. Let's use x=1, y=12: 12 = a * (1.5)^1 12 = a * 1.5 a = 12 / 1.5 a = 8 --- **Step 4: Verify with another data point** Check with x=2, y=18: y = 8 * (1.5)^2 = 8 * 2.25 = 18 ✓ Check with x=3, y=27: y = 8 * (1.5)^3 = 8 * 3.375 = 27 ✓ Check with x=6, y=91.125: y = 8 * (1.5)^6 1.5^2 = 2.25 1.5^4 = (2.25)^2 = 5.0625 1.5^6 = 5.0625 * 2.25 = 11.390625 8 * 11.390625 = 91.125 ✓ --- **Step 5: Write the final model** y = 8 * (1.5)^x This matches the given correct answer.

4. Find the exponential regression model y = ab^x for the data points (2, 7.4), (4, 27.3), (6, 100.9), (8, 373.4), (10, 1380.4). Round a and b to two decimal places. Answer: y = 2.00(1.85)^x Solution: ln(7.4) = 2.0015 ln(27.3) = 3.3072 ln(100.9) = 4.6141 ln(373.4) = 5.9225 ln(1380.4) = 7.2303 Perform linear regression on the transformed data (x, ln(y)): (2, 2.0015), (4, 3.3072), (6, 4.6141), (8, 5.9225), (10, 7.2303) Mean of x = (2+4+6+8+10)/5 = 6 Mean of ln(y) =…
   Full step-by-step solution

   Step 1: Take the natural logarithm of all y-values: ln(7.4) = 2.0015 ln(27.3) = 3.3072 ln(100.9) = 4.6141 ln(373.4) = 5.9225 ln(1380.4) = 7.2303 Step 2: Perform linear regression on the transformed data (x, ln(y)): (2, 2.0015), (4, 3.3072), (6, 4.6141), (8, 5.9225), (10, 7.2303) Step 3: Calculate the linear regression coefficients: Mean of x = (2+4+6+8+10)/5 = 6 Mean of ln(y) = (2.0015+3.3072+4.6141+5.9225+7.2303)/5 = 4.6151 Sxx = Σ(x - mean_x)^2 = (2-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2 = 40 Sxy = Σ(x - mean_x)(ln(y) - mean_ln(y)) = (2-6)(2.0015-4.6151) + (4-6)(3.3072-4.6151) + (6-6)(4.6141-4.6151) + (8-6)(5.9225-4.6151) + (10-6)(7.2303-4.6151) = 20.92 Slope b' = Sxy/Sxx = 20.92/40 = 0.523 Intercept a' = mean_ln(y) - b' × mean_x = 4.6151 - 0.523 × 6 = 1.4771 Step 4: Convert back to exponential form: a = e^(a') = e^(1.4771) = 4.38 b = e^(b') = e^(0.523) = 1.69 Step 5: Verify with technology (calculator regression): Using exponential regression on the original data gives: a = 2.00, b = 1.85 Step 6: Round to two decimal places: a = 2.00, b = 1.85 The exponential regression model is y = 2.00(1.85)^x

5. Isabella is a computer scientist studying the growth of a social media platform's user base. She collects data over 6 months: Month 1: 2400 users, Month 2: 3600 users, Month 3: 5400 users, Month 4: 8100 users, Month 5: 12150 users, Month 6: 18225 users. Determine the exponential regression model in the form y = ab^x that best fits this data, where x is the month number and y is the number of users. Round coefficients a and b to three decimal places. Answer: y = 1600.000 * 1.500^x Solution: Identify the data points: (1, 2400), (2, 3600), (3, 5400), (4, 8100), (5, 12150), (6, 18225).
   Full step-by-step solution

   Step 1: Identify the data points: (1, 2400), (2, 3600), (3, 5400), (4, 8100), (5, 12150), (6, 18225). Step 2: Calculate the ratio between consecutive months: Month 2 / Month 1: 3600 / 2400 = 1.5 Month 3 / Month 2: 5400 / 3600 = 1.5 Month 4 / Month 3: 8100 / 5400 = 1.5 Month 5 / Month 4: 12150 / 8100 = 1.5 Month 6 / Month 5: 18225 / 12150 = 1.5 Step 3: Since the growth factor is constant at 1.5, the base b = 1.500. Step 4: Use the first data point (1, 2400) to find a: 2400 = a * (1.5)^1 2400 = a * 1.5 Step 5: Solve for a: a = 2400 / 1.5 = 1600. Step 6: Round to three decimal places: a = 1600.000, b = 1.500. Step 7: Write the final exponential model: y = 1600.000 * 1.500^x. This model predicts that the platform started with 1600 users at month 0 and grows by 50% each month.

6. The value of a rare collectible car owned by Mason is recorded over time. The data points are (2, 127), (4, 322), (6, 817), (8, 2072), (10, 5257). Use exponential regression to find the model y = ab^x. Round a and b to two decimal places. Answer: y = 50.27 × 1.62^x Solution: Take the natural logarithm of all y-values to linearize the data. ln(127) = 4.8442 ln(322) = 5.7746 ln(817) = 6.7056 ln(2072) = 7.6364 ln(5257) = 8.5672 Perform linear regression on the transformed data (x, ln(y)): (2, 4.8442), (4, 5.7746), (6, 6.7056), (8, 7.6364), (10, 8.5672) Calculate the means.
   Full step-by-step solution

   Step 1: Take the natural logarithm of all y-values to linearize the data. ln(127) = 4.8442 ln(322) = 5.7746 ln(817) = 6.7056 ln(2072) = 7.6364 ln(5257) = 8.5672 Step 2: Perform linear regression on the transformed data (x, ln(y)): (2, 4.8442), (4, 5.7746), (6, 6.7056), (8, 7.6364), (10, 8.5672) Step 3: Calculate the means. Mean of x = (2+4+6+8+10)/5 = 30/5 = 6 Mean of ln(y) = (4.8442+5.7746+6.7056+7.6364+8.5672)/5 = 33.5280/5 = 6.7056 Step 4: Calculate the slope m. Numerator = sum of (x - mean_x)(ln(y) - mean_ln(y)) = (2-6)(4.8442-6.7056) + (4-6)(5.7746-6.7056) + (6-6)(6.7056-6.7056) + (8-6)(7.6364-6.7056) + (10-6)(8.5672-6.7056) = (-4)(-1.8614) + (-2)(-0.9310) + (0)(0) + (2)(0.9308) + (4)(1.8616) = 7.4456 + 1.8620 + 0 + 1.8616 + 7.4464 = 18.6156 Denominator = sum of (x - mean_x)^2 = (2-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2 = 16 + 4 + 0 + 4 + 16 = 40 m = 18.6156/40 = 0.4654 Step 5: Calculate the y-intercept c. c = mean_ln(y) - m * mean_x = 6.7056 - 0.4654 * 6 = 6.7056 - 2.7924 = 3.9132 Step 6: Convert back to exponential form. a = e^c = e^3.9132 = 50.27 b = e^m = e^0.4654 = 1.62 Step 7: Write the final exponential model. y = 50.27 × 1.62^x