Exponential Models

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Answer Key & Explanations

Exponential Models · Grade 11 · Worksheet 1

1. Isabella collected data on the cooling of a liquid: time (min) = 0, 8, 16, 24; temperature (°C) = 95, 47.5, 23.75, 11.875. Is an exponential model appropriate? Justify your answer. Answer: B. yes Solution: Check the time intervals: 8-0 = 8, 16-8 = 8, 24-16 = 8. Time increases by constant intervals of 8 minutes.
   Full step-by-step solution

   Step 1: Check the time intervals: 8-0 = 8, 16-8 = 8, 24-16 = 8. Time increases by constant intervals of 8 minutes. Step 2: Check the ratios between consecutive temperature values: 47.5 ÷ 95 = 0.5 23.75 ÷ 47.5 = 0.5 11.875 ÷ 23.75 = 0.5 Step 3: Since the ratio between consecutive temperature values is constant at 0.5, this indicates exponential decay. Step 4: Therefore, an exponential model is appropriate for this data.

2. 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)

3. A biologist is studying bacterial growth in a lab culture. The initial population is 500 bacteria, and observations show the population doubles every 3 hours. The biologist models this growth with the function P(t) = P₀ * e^(kt), where t is time in hours. Determine the value of the continuous growth rate k for this bacterial population. Answer: ln(2)/3 Solution: We are given that the initial population is 500 bacteria and that the population doubles every 3 hours. P(t) = P₀ * e^(k t) where P₀ = 500, t is in hours, and k is the continuous growth rate we need to find.
   Full step-by-step solution

   We are given that the initial population is 500 bacteria and that the population doubles every 3 hours. The model is: P(t) = P₀ * e^(k t) where P₀ = 500, t is in hours, and k is the continuous growth rate we need to find. --- **Step 1: Use the doubling time information.** If the population doubles every 3 hours, then: P(3) = 2 * P₀ Substitute into the model: 500 * e^(k * 3) = 2 * 500 --- **Step 2: Simplify the equation.** Divide both sides by 500: e^(3k) = 2 --- **Step 3: Solve for k.** Take the natural logarithm of both sides: ln(e^(3k)) = ln(2) 3k = ln(2) Divide both sides by 3: k = ln(2) / 3 --- **Final Answer:** k = ln(2)/3

4. A research scientist is studying bacterial growth in a lab culture. The initial population is 500 bacteria, and the population doubles every 3 hours. The scientist models this growth with the function P(t) = P₀ × 2^(t/k), where P(t) is the population at time t hours, P₀ is the initial population, and k is the doubling time. Determine how many bacteria will be present after 9 hours of growth. Answer: 4000 Solution: Identify the given values from the problem. Initial population \( P_0 = 500 \) bacteria. Doubling time \( k = 3 \) hours.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the given values from the problem.** Initial population \( P_0 = 500 \) bacteria. Doubling time \( k = 3 \) hours. Time elapsed \( t = 9 \) hours. Growth model: \( P(t) = P_0 \times 2^{(t/k)} \). --- **Step 2: Substitute the known values into the formula.** \[ P(9) = 500 \times 2^{(9/3)} \] --- **Step 3: Simplify the exponent.** \[ 9/3 = 3 \] So: \[ P(9) = 500 \times 2^{3} \] --- **Step 4: Calculate \( 2^{3} \).** \[ 2^{3} = 8 \] So: \[ P(9) = 500 \times 8 \] --- **Step 5: Multiply to find the final population.** \[ 500 \times 8 = 4000 \] --- **Step 6: Interpret the result.** After 9 hours, the population will be 4000 bacteria. --- **Final Answer:** 4000

5. Noah is analyzing the growth of a bacteria culture in a laboratory. He records the population (in thousands) every 6 hours and obtains the following data: Time (hours): 0 6 12 18 24 Population (thousands): 1.6 4.8 14.4 43.2 129.6 Does this data suggest that the population growth follows a linear model or an exponential model? Justify your answer by explaining the pattern in the data. Answer: Exponential model Solution: Observe that time increases by a constant 6 hours between each pair of data points. Calculate the differences between consecutive populations: 4.8 - 1.6 = 3.2, 14.4 - 4.8 = 9.6, 43.2 - 14.4 = 28.8, 129.6 - 43.2 = 86.4.
   Full step-by-step solution

   Step 1: Observe that time increases by a constant 6 hours between each pair of data points. Step 2: Calculate the differences between consecutive populations: 4.8 - 1.6 = 3.2, 14.4 - 4.8 = 9.6, 43.2 - 14.4 = 28.8, 129.6 - 43.2 = 86.4. These differences are not constant (3.2, 9.6, 28.8, 86.4), so the model is not linear. Step 3: Calculate the ratios of each population to the previous one: 4.8 / 1.6 = 3, 14.4 / 4.8 = 3, 43.2 / 14.4 = 3, 129.6 / 43.2 = 3. The ratio is constant at 3. Step 4: Since the ratio is constant (each population is multiplied by 3 every 6 hours), the growth follows an exponential model. The answer is exponential model.

6. Matiu measured the temperature of a cooling liquid every 5 minutes: t=0: 256°C, t=5: 128°C, t=10: 64°C, t=15: 32°C. Is an exponential model appropriate? Justify your answer by calculating the ratio between consecutive temperature values. Answer: A. yes Solution: Calculate the ratio between consecutive temperature values From t=0 to t=5: 128 ÷ 256 = 0.5 From t=5 to t=10: 64 ÷ 128 = 0.5 From t=10 to t=15: 32 ÷ 64 = 0.5 All three ratios equal 0.5, which is constant.
   Full step-by-step solution

   Step 1: Calculate the ratio between consecutive temperature values From t=0 to t=5: 128 ÷ 256 = 0.5 From t=5 to t=10: 64 ÷ 128 = 0.5 From t=10 to t=15: 32 ÷ 64 = 0.5 Step 2: Analyze the ratios All three ratios equal 0.5, which is constant. Step 3: Determine if exponential model is appropriate Since the ratio between consecutive y-values is constant when x increases by a constant amount (5 minutes), an exponential model is appropriate. Conclusion: Yes, an exponential model is appropriate because the temperature halves every 5 minutes, showing a constant multiplicative rate of change.

7. A right triangle is positioned on a coordinate plane with vertices at (0,0), (6,0), and (0,8). A circle is inscribed in this triangle, tangent to all three sides. What is the radius of this inscribed circle? Answer: 2 Solution: Identify the triangle's side lengths. The vertices are (0,0), (6,0), and (0,8). The side along the x-axis has length 6.
   Full step-by-step solution

   Step 1: Identify the triangle's side lengths. The vertices are (0,0), (6,0), and (0,8). The side along the x-axis has length 6. The side along the y-axis has length 8. The hypotenuse length is sqrt((6-0)^2 + (0-8)^2) = sqrt(36 + 64) = sqrt(100) = 10. Step 2: Calculate the triangle's area. Area = (1/2) * base * height = (1/2) * 6 * 8 = 24 square units. Step 3: Calculate the triangle's semiperimeter (s). Perimeter = 6 + 8 + 10 = 24. Semiperimeter s = 24 / 2 = 12. Step 4: Use the formula relating area, semiperimeter, and inradius (r): Area = r * s. 24 = r * 12 Step 5: Solve for r. r = 24 / 12 = 2. The radius of the inscribed circle is 2.