Exponential Parameters

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Answer Key & Explanations

Exponential Parameters · Grade 11 · Worksheet 1

1. A biologist is studying bacterial growth in a lab culture. The initial population is 500 bacteria, and it doubles every 3 hours. The growth can be modeled by the function P(t) = P₀ * e^(kt), where t is time in hours. Determine the value of the continuous growth rate constant k, and then calculate the population after 9 hours. Answer: 4000 Solution: Initial population P₀ = 500. Doubling time = 3 hours. Growth model: P(t) = P₀ * e^(k t).
   Full step-by-step solution

   Step 1: Understand the problem and the given information. Initial population P₀ = 500. Doubling time = 3 hours. Growth model: P(t) = P₀ * e^(k t). We need to find k first, then find P(9). --- Step 2: Use the doubling time to find k. If the population doubles in 3 hours, then P(3) = 2 * P₀. So: 500 * e^(k * 3) = 2 * 500 Divide both sides by 500: e^(3k) = 2 --- Step 3: Solve for k. Take natural logarithm of both sides: ln(e^(3k)) = ln(2) 3k = ln(2) k = ln(2) / 3 Numerically, ln(2) ≈ 0.693147, so: k ≈ 0.693147 / 3 ≈ 0.231049 --- Step 4: Write the specific growth function. P(t) = 500 * e^(k t) P(t) = 500 * e^( (ln(2)/3) * t ) --- Step 5: Find the population after 9 hours. Method 1: Using doubling times. Doubling time is 3 hours, so in 9 hours there are 9 / 3 = 3 doublings. Initial: 500 After 3 hours: 1000 After 6 hours: 2000 After 9 hours: 4000 Method 2: Using the formula. P(9) = 500 * e^( (ln(2)/3) * 9 ) = 500 * e^( ln(2) * 3 ) = 500 * e^( ln(8) ) = 500 * 8 = 4000 --- Step 6: Conclusion. The continuous growth rate constant k = ln(2)/3. The population after 9 hours is 4000 bacteria. Final answer: 4000

2. Isabella's investment grows according to I(t) = 1200(1.072)^t, where t is years. What does 1200 represent? What does 1.072 represent? Answer: 1200 represents the initial investment amount, 1.072 represents the annual growth factor Solution: In the exponential function I(t) = 1200(1.072)^t, the parameter 'a' represents the initial value when t = 0. When t = 0, I(0) = 1200(1.072)^0 = 1200 × 1 = 1200, so 1200 is Isabella's initial investment.
   Full step-by-step solution

   Step 1: In the exponential function I(t) = 1200(1.072)^t, the parameter 'a' represents the initial value when t = 0. Step 2: When t = 0, I(0) = 1200(1.072)^0 = 1200 × 1 = 1200, so 1200 is Isabella's initial investment. Step 3: The parameter 'b' represents the growth factor per time period. Since b = 1.072 > 1, this indicates growth. Step 4: The growth rate is b - 1 = 1.072 - 1 = 0.072, or 7.2% per year. Step 5: Therefore, 1200 represents the initial investment amount and 1.072 represents the annual growth factor.

3. Sophia's investment grows according to V(t) = 1600(1.06)^t, where t is time in years. What does the number 1600 represent? What does the number 1.06 represent? Answer: 1600 represents the initial investment amount in dollars, and 1.06 represents the annual growth factor, meaning the investment grows by 6% each year. Solution: Identify the exponential function form: V(t) = a·b^t Compare to given function: V(t) = 1600(1.06)^t The parameter 'a' = 1600 represents the initial value when t = 0 When t = 0: V(0) = 1600(1.06)^0 = 1600(1) = 1600 The parameter 'b' = 1.06 represents the growth factor Since b > 1, this indicates…
   Full step-by-step solution

   Step 1: Identify the exponential function form: V(t) = a·b^t Step 2: Compare to given function: V(t) = 1600(1.06)^t Step 3: The parameter 'a' = 1600 represents the initial value when t = 0 Step 4: When t = 0: V(0) = 1600(1.06)^0 = 1600(1) = 1600 Step 5: The parameter 'b' = 1.06 represents the growth factor Step 6: Since b > 1, this indicates growth Step 7: The growth rate is b - 1 = 1.06 - 1 = 0.06 = 6% Step 8: Therefore, 1600 is the initial investment and 1.06 means 6% annual growth

4. Hana is monitoring the growth of a rare fern in a controlled environment. The height of the fern, in centimeters, is modeled by the exponential function H(t) = 8(1.06)^t, where t is the number of weeks since the fern was first planted. Interpret the meaning of the parameters 8 and 1.06 in this context. Answer: The parameter 8 represents the initial height of the fern in centimeters when it was first planted (t = 0). The parameter 1.06 represents the growth factor per week, meaning the fern's height increases by 6% each week. Solution: Identify the general form of an exponential function: H(t) = a * b^t, where a is the initial value and b is the growth or decay factor. In the given function H(t) = 8(1.06)^t, a = 8 and b = 1.06.
   Full step-by-step solution

   Step 1: Identify the general form of an exponential function: H(t) = a * b^t, where a is the initial value and b is the growth or decay factor. Step 2: In the given function H(t) = 8(1.06)^t, a = 8 and b = 1.06. Step 3: Interpret a = 8: When t = 0, H(0) = 8(1.06)^0 = 8 * 1 = 8. So the initial height of the fern when it was first planted is 8 centimeters. Step 4: Interpret b = 1.06: Since 1.06 > 1, this indicates growth. The factor 1.06 means the height multiplies by 1.06 each week. A factor of 1.06 corresponds to a 6% increase per week (because 1.06 = 1 + 0.06, and 0.06 = 6/100 = 6%). Step 5: Verify: After 1 week, H(1) = 8 * 1.06 = 8.48 cm, which is 8 + 0.48 cm more than the initial height. 0.48/8 = 0.06 = 6% increase. The answer: 8 is the initial height in centimeters; 1.06 is the weekly growth factor representing a 6% increase per week.

5. A radioactive substance decays exponentially, and its mass is modeled by the function M(t) = 80e^(-0.05t), where M is the mass in grams and t is time in years. On a coordinate plane, the graph of this function shows the mass decreasing over time. What is the half-life of this substance? (Round your answer to the nearest tenth of a year.) Answer: 13.9 Solution: M(t) = 80 e^(-0.05 t) Half-life is the time t such that the mass is half of the initial mass. Initial mass at t = 0: M(0) = 80 e^(0) = 80 grams. Half of that is 40 grams.
   Full step-by-step solution

   We are given the exponential decay model: M(t) = 80 e^(-0.05 t) --- **Step 1: Understand half-life definition** Half-life is the time t such that the mass is half of the initial mass. Initial mass at t = 0: M(0) = 80 e^(0) = 80 grams. Half of that is 40 grams. So we solve: 80 e^(-0.05 t) = 40 --- **Step 2: Divide both sides by 80** e^(-0.05 t) = 40/80 e^(-0.05 t) = 1/2 --- **Step 3: Take natural logarithm of both sides** ln(e^(-0.05 t)) = ln(1/2) -0.05 t = ln(1/2) --- **Step 4: Simplify ln(1/2)** ln(1/2) = -ln(2) So: -0.05 t = -ln(2) --- **Step 5: Cancel the negative signs** 0.05 t = ln(2) --- **Step 6: Solve for t** t = ln(2) / 0.05 --- **Step 7: Compute numerically** ln(2) ≈ 0.693147 t ≈ 0.693147 / 0.05 t ≈ 13.86294 --- **Step 8: Round to nearest tenth** t ≈ 13.9 years --- **Final answer:** 13.9

6. A pharmaceutical company is testing a new drug that is metabolized by the body following an exponential decay model. The concentration of the drug in the bloodstream is given by C(t) = 120 × e^(-0.15t), where C is measured in milligrams per liter and t is time in hours. Determine how many hours it will take for the drug concentration to decrease to 30 mg/L. Answer: 9.24 Solution: Set up the equation using the given function: 30 = 120 × e^(-0.15t) Divide both sides by 120: 30/120 = e^(-0.15t) Simplify: 0.25 = e^(-0.15t) Take the natural logarithm of both sides: ln(0.25) = ln(e^(-0.15t)) Simplify using logarithm properties: ln(0.25) = -0.15t Calculate ln(0.25): ln(0.25) ≈…
   Full step-by-step solution

   Step 1: Set up the equation using the given function: 30 = 120 × e^(-0.15t) Step 2: Divide both sides by 120: 30/120 = e^(-0.15t) Step 3: Simplify: 0.25 = e^(-0.15t) Step 4: Take the natural logarithm of both sides: ln(0.25) = ln(e^(-0.15t)) Step 5: Simplify using logarithm properties: ln(0.25) = -0.15t Step 6: Calculate ln(0.25): ln(0.25) ≈ -1.3863 Step 7: Substitute: -1.3863 = -0.15t Step 8: Solve for t: t = -1.3863 / -0.15 Step 9: Calculate: t ≈ 9.242 Step 10: Round to two decimal places: t ≈ 9.24 hours The answer is 9.24.

7. Emma is studying a radioactive substance that decays according to the function M(t) = 800(0.85)^t, where M is the mass in grams and t is time in years. What does the value 800 represent in this context? What does the value 0.85 represent? Answer: 800 represents the initial mass of the substance in grams; 0.85 represents the decay factor per year, meaning the mass decreases by 15% each year Solution: Identify the general form of an exponential function: f(t) = a·b^t In this function M(t) = 800(0.85)^t, we have a = 800 and b = 0.85 When t = 0, M(0) = 800(0.85)^0 = 800(1) = 800 grams This means 800 represents the initial mass of the radioactive substance The base b = 0.85 represents the decay…
   Full step-by-step solution

   Step 1: Identify the general form of an exponential function: f(t) = a·b^t Step 2: In this function M(t) = 800(0.85)^t, we have a = 800 and b = 0.85 Step 3: When t = 0, M(0) = 800(0.85)^0 = 800(1) = 800 grams Step 4: This means 800 represents the initial mass of the radioactive substance Step 5: The base b = 0.85 represents the decay factor per year Step 6: Since 0.85 = 85%, this means each year the mass becomes 85% of what it was the previous year Step 7: This corresponds to a 15% decrease per year (100% - 85% = 15%) Step 8: Therefore, 800 represents the initial mass in grams, and 0.85 represents the decay factor per year, indicating a 15% annual decrease in mass.

8. Matiu's investment grows according to V(t) = 8500(1.042)^t, where t is in years. What does 8500 represent? What does 1.042 represent? Answer: 8500 is the initial investment amount, 1.042 is the annual growth factor Solution: The function is V(t) = 8500(1.042)^t When t = 0 (initial time), V(0) = 8500(1.042)^0 = 8500 × 1 = 8500 Therefore, 8500 represents the initial value of the investment The base 1.042 represents the growth factor per time period (year) Since 1.042 > 1, this indicates growth, and the investment…
   Full step-by-step solution

   Step 1: The function is V(t) = 8500(1.042)^t Step 2: When t = 0 (initial time), V(0) = 8500(1.042)^0 = 8500 × 1 = 8500 Step 3: Therefore, 8500 represents the initial value of the investment Step 4: The base 1.042 represents the growth factor per time period (year) Step 5: Since 1.042 > 1, this indicates growth, and the investment grows by 4.2% each year Step 6: 8500 is the initial investment amount, 1.042 is the annual growth factor