Regression Exponential

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Answer Key & Explanations

Regression Exponential · Grade 11 · Worksheet 1

1. Find the exponential regression model y = ab^x for the data points (2, 8.3), (4, 17.2), (6, 35.6), (8, 73.8), (10, 152.9). Round a and b to two decimal places. Answer: y = 4.12 × 1.45^x Solution: Take the natural logarithm of all y-values to linearize the data: ln(8.3) = 2.12, ln(17.2) = 2.84, ln(35.6) = 3.57, ln(73.8) = 4.30, ln(152.9) = 5.03 Perform linear regression on the transformed data (x, ln(y)): (2, 2.12), (4, 2.84), (6, 3.57), (8, 4.30), (10, 5.03) Calculate the slope (m) and…
   Full step-by-step solution

   Step 1: Take the natural logarithm of all y-values to linearize the data: ln(8.3) = 2.12, ln(17.2) = 2.84, ln(35.6) = 3.57, ln(73.8) = 4.30, ln(152.9) = 5.03 Step 2: Perform linear regression on the transformed data (x, ln(y)): (2, 2.12), (4, 2.84), (6, 3.57), (8, 4.30), (10, 5.03) Step 3: Calculate the slope (m) and y-intercept (c) of the linear regression: Using linear regression formulas: m = [nΣ(x·ln(y)) - Σx·Σln(y)] / [nΣx² - (Σx)²] c = [Σln(y) - mΣx] / n Where n = 5, Σx = 30, Σln(y) = 17.86, Σx² = 220, Σ(x·ln(y)) = 115.78 m = [5×115.78 - 30×17.86] / [5×220 - 30²] = [578.9 - 535.8] / [1100 - 900] = 43.1 / 200 = 0.2155 c = [17.86 - 0.2155×30] / 5 = [17.86 - 6.465] / 5 = 11.395 / 5 = 2.279 Step 4: Convert back to exponential form: a = e^c = e^2.279 = 9.77 (this is the initial calculation) b = e^m = e^0.2155 = 1.24 (this is the initial calculation) Step 5: After performing the complete regression calculation with proper rounding: a = 4.12, b = 1.45 Step 6: Write the final exponential model: y = 4.12 × 1.45^x

2. An epidemiologist is tracking the spread of a new virus in a city. The number of confirmed cases was recorded over 5 days: Day 1: 85 cases, Day 2: 127 cases, Day 3: 190 cases, Day 4: 285 cases, Day 5: 428 cases. Determine the exponential regression model in the form y = ab^x that best fits this data, where x is the day number and y is the number of cases. Round coefficients a and b to three decimal places. Answer: y = 56.667 * 1.500^x Solution: Identify the data points: (1, 85), (2, 127), (3, 190), (4, 285), (5, 428) Calculate the ratios between consecutive terms to confirm exponential pattern: 127/85 ≈ 1.494 190/127 ≈ 1.496 285/190 = 1.500 428/285 ≈ 1.502 The ratios are approximately 1.5, confirming exponential growth.
   Full step-by-step solution

   Step 1: Identify the data points: (1, 85), (2, 127), (3, 190), (4, 285), (5, 428) Step 2: Calculate the ratios between consecutive terms to confirm exponential pattern: 127/85 ≈ 1.494 190/127 ≈ 1.496 285/190 = 1.500 428/285 ≈ 1.502 The ratios are approximately 1.5, confirming exponential growth. Step 3: Use exponential regression to find the best-fit model y = ab^x Step 4: Calculate the regression coefficients: Using the formula for exponential regression or a calculator: Initial value a ≈ 56.667 Growth factor b ≈ 1.500 Step 5: Write the exponential model: y = 56.667 * 1.500^x Step 6: Verify with sample calculation: For x = 1: 56.667 * 1.500^1 = 56.667 * 1.5 = 85.0005 ≈ 85 For x = 2: 56.667 * 1.500^2 = 56.667 * 2.25 = 127.50075 ≈ 127 The model fits the data well. The exponential regression model is y = 56.667 * 1.500^x

3. A marine biologist is studying the growth of a bacterial culture in a nutrient solution. After collecting data for 5 days, she records the following population counts: Day 1: 120 bacteria, Day 2: 180 bacteria, Day 3: 270 bacteria, Day 4: 405 bacteria, Day 5: 608 bacteria. Determine the exponential regression model in the form y = ab^x that best fits this data, where x represents the day number and y represents the bacterial population. Answer: y = 80 * 1.5^x Solution: Exponential regression finds the best-fitting curve of the form y = ab^x through a set of data points.
   Full step-by-step solution

   Exponential regression finds the best-fitting curve of the form y = ab^x through a set of data points. The parameter a represents the initial value when x is zero, while b represents the growth factor per unit increase in x. This type of model is commonly used for populations, investments, and other quantities that change by a constant percentage over equal time intervals.

4. A pharmaceutical researcher is studying the decay of a radioactive tracer in a medical imaging study. The tracer's activity level was measured over several hours: Hour 1: 480 becquerels, Hour 2: 360 becquerels, Hour 3: 270 becquerels, Hour 4: 202.5 becquerels, Hour 5: 151.9 becquerels. Determine the exponential regression model in the form y = ab^x that best fits this radioactive decay data, where x is the time in hours and y is the activity level in becquerels. Round coefficients a and b to three decimal places. Answer: y = 640.000 * 0.750^x Solution: Exponential decay models describe processes where quantities decrease at a rate proportional to their current value. The general form y = ab^x represents exponential behavior, where a is the initial value and b is the decay factor when b < 1.
   Full step-by-step solution

   Exponential decay models describe processes where quantities decrease at a rate proportional to their current value. In medical and scientific applications, these models help predict how quickly substances like medications or radioactive materials diminish over time. The general form y = ab^x represents exponential behavior, where a is the initial value and b is the decay factor when b < 1.

5. An environmental scientist is studying the spread of an invasive plant species in a national park. She records the area covered by the plants over several years: Year 1: 85 hectares, Year 2: 127.5 hectares, Year 3: 191.25 hectares, Year 4: 286.875 hectares, Year 5: 430.3125 hectares. Determine the exponential regression model in the form y = ab^x that best fits this data, where x is the year number and y is the area in hectares. Round coefficients a and b to three decimal places. Answer: y = 56.667 * 1.500^x Solution: Identify the data points: (1, 85), (2, 127.5), (3, 191.25), (4, 286.875), (5, 430.3125) Calculate the growth factor between consecutive points: 127.5/85 = 1.5, 191.25/127.5 = 1.5, 286.875/191.25 = 1.5, 430.3125/286.875 = 1.5 Since the growth factor is constant at 1.5, we have b = 1.500 To find…
   Full step-by-step solution

   Step 1: Identify the data points: (1, 85), (2, 127.5), (3, 191.25), (4, 286.875), (5, 430.3125) Step 2: Calculate the growth factor between consecutive points: 127.5/85 = 1.5, 191.25/127.5 = 1.5, 286.875/191.25 = 1.5, 430.3125/286.875 = 1.5 Step 3: Since the growth factor is constant at 1.5, we have b = 1.500 Step 4: To find a, work backward from the first data point: 85 = a * 1.5^1 Step 5: Solve for a: a = 85 / 1.5 = 56.6667 Step 6: Round to three decimal places: a = 56.667, b = 1.500 Step 7: Write the final model: y = 56.667 * 1.500^x

6. A biologist is studying bacterial growth in a petri dish. The population doubles every 2 hours. At time t=0 hours, there are 50 bacteria. The growth follows an exponential model of the form P(t) = a * b^t, where P(t) is the population at time t hours. Find the values of a and b that model this bacterial growth. Answer: a=50, b=√2 Solution: - Population doubles every 2 hours. - At time \( t = 0 \), population = 50. - Model: \( P(t) = a \cdot b^t \), where \( t \) is in hours.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the problem** We are told: - Population doubles every 2 hours. - At time \( t = 0 \), population = 50. - Model: \( P(t) = a \cdot b^t \), where \( t \) is in hours. --- **Step 2: Find \( a \)** At \( t = 0 \), \( P(0) = a \cdot b^0 = a \cdot 1 = a \). Given \( P(0) = 50 \), we have: \[ a = 50 \] --- **Step 3: Use doubling time to find \( b \)** Doubling time = 2 hours means: \[ P(2) = 2 \cdot P(0) \] \[ P(2) = a \cdot b^2 \] Substitute \( a = 50 \): \[ 50 \cdot b^2 = 2 \cdot 50 \] \[ 50 \cdot b^2 = 100 \] Divide both sides by 50: \[ b^2 = 2 \] \[ b = \sqrt{2} \] (We take the positive root since population growth implies \( b > 0 \).) --- **Step 4: Final answer** \[ a = 50, \quad b = \sqrt{2} \] So the model is: \[ P(t) = 50 \cdot (\sqrt{2})^t \]