Exponential Growth and Decay

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Answer Key & Explanations

Exponential Growth and Decay · Grade 10 · Worksheet 2

1. Emma is studying a population of bacteria in a Petri dish. At 10:00 AM, she observes 500 bacteria. By 2:00 PM, the population has grown to 8000 bacteria. The growth is exponential and follows the model P(t) = P₀ · b^t, where t is the number of hours after 10:00 AM. What is the value of the growth factor b (rounded to two decimal places)? Answer: 2.00 Solution: Identify the given values. Initial population P₀ = 500 at t = 0 (10:00 AM). At t = 4 hours (2:00 PM), P(4) = 8000.
   Full step-by-step solution

   Step 1: Identify the given values. Initial population P₀ = 500 at t = 0 (10:00 AM). At t = 4 hours (2:00 PM), P(4) = 8000. Step 2: Write the exponential equation: P(t) = 500 · b^t. Step 3: Substitute the known point: 8000 = 500 · b^4. Step 4: Divide both sides by 500: 8000 / 500 = b^4 → 16 = b^4. Step 5: Solve for b by taking the fourth root: b = 16^(1/4) = 2. Step 6: Rounded to two decimal places, b = 2.00. The growth factor is 2.00.

2. A sample of a radioactive isotope has an initial mass of 96 grams. Its half-life is 9 years. Write an exponential decay model A(t) for the mass remaining after t years, and use it to find the mass remaining after 27 years. Answer: 12 Solution: The initial mass is A₀ = 96 grams. The half-life is 9 years. The exponential decay model is A(t) = 96(0.5)^(t/9).
   Full step-by-step solution

   Step 1: The initial mass is A₀ = 96 grams. The half-life is 9 years. The exponential decay model is A(t) = 96(0.5)^(t/9). Step 2: Substitute t = 27: A(27) = 96(0.5)^(27/9). Step 3: Simplify the exponent: 27/9 = 3, so A(27) = 96(0.5)^3. Step 4: Calculate (0.5)^3 = 0.125. Step 5: Multiply: 96 × 0.125 = 12. The mass remaining after 27 years is 12 grams.

3. A right circular cone has a height of 12 cm and a base radius of 5 cm. The cone is intersected by a plane parallel to its base, creating a smaller cone at the top with height 4 cm. What is the volume of the frustum (the remaining portion after removing the smaller top cone)? Answer: 700π/3 cm³ Solution: When a cone is cut by a plane parallel to its base, the resulting smaller cone is similar to the original cone. This similarity allows us to find the radius of the smaller cone using proportional relationships. The volume of a cone is calculated using the formula V = (1/3)πr²h.
   Full step-by-step solution

   When a cone is cut by a plane parallel to its base, the resulting smaller cone is similar to the original cone. This similarity allows us to find the radius of the smaller cone using proportional relationships. The volume of a cone is calculated using the formula V = (1/3)πr²h. For a frustum, we subtract the volume of the smaller cone from the volume of the larger cone to find the volume of the remaining solid.

4. Isabella invests $2,700 in a savings account that earns 7% annual interest, compounded yearly. Write the exponential function that models the account balance after t years. Answer: A(t) = 2700(1.07)^t Solution: Identify the initial amount (principal). Isabella invests $2,700, so a = 2700. Identify the annual growth rate.
   Full step-by-step solution

   Step 1: Identify the initial amount (principal). Isabella invests $2,700, so a = 2700. Step 2: Identify the annual growth rate. The interest rate is 7%, which as a decimal is 0.07. Step 3: Determine the growth factor. For exponential growth, the growth factor b = 1 + r = 1 + 0.07 = 1.07. Step 4: Write the exponential function. The general form is A(t) = a * b^t, where t is time in years. Substituting the values: A(t) = 2700 * (1.07)^t. Step 5: The final answer is A(t) = 2700(1.07)^t.

5. Matiu is analyzing the growth of a rare tree fern in a protected forest reserve. The current population of fern plants is 2,400. Due to ideal conditions, the population is expected to increase by 8% every 2 years. Using the exponential growth model P(t) = P₀ × (1 + r)^(t/k), where P₀ is the initial population, r is the growth rate per period, t is time in years, and k is the number of years per growth period, determine the predicted fern population after 6 years. Round your answer to the nearest whole number. Answer: 3023 Solution: Identify the given values. Initial population P₀ = 2400 Growth rate per period r = 8% = 0.08 Years per growth period k = 2 years Total time t = 6 years Determine the number of growth periods.
   Full step-by-step solution

   Step 1: Identify the given values. Initial population P₀ = 2400 Growth rate per period r = 8% = 0.08 Years per growth period k = 2 years Total time t = 6 years Step 2: Determine the number of growth periods. Number of periods = t / k = 6 / 2 = 3 Step 3: Apply the exponential growth formula. P(t) = P₀ × (1 + r)^(t/k) P(6) = 2400 × (1 + 0.08)^3 P(6) = 2400 × (1.08)^3 Step 4: Calculate (1.08)^3. 1.08^2 = 1.1664 1.1664 × 1.08 = 1.259712 Step 5: Multiply by the initial population. P(6) = 2400 × 1.259712 P(6) = 3023.3088 Step 6: Round to the nearest whole number. 3023.3088 rounds to 3023 The answer is 3023.

6. Liam invests $3,500 in a savings account that earns 9% annual interest compounded annually. Write the exponential function A(t) that models the account balance after t years, and determine the balance after 7 years, rounded to the nearest dollar. Answer: 6398 Solution: The initial amount is a = 3500. The annual interest rate is 9% = 0.09, so the growth factor is b = 1 + 0.09 = 1.09. The exponential growth model is A(t) = 3500(1.09)^t.
   Full step-by-step solution

   Step 1: The initial amount is a = 3500. The annual interest rate is 9% = 0.09, so the growth factor is b = 1 + 0.09 = 1.09. The exponential growth model is A(t) = 3500(1.09)^t. Step 2: To find the balance after 7 years, substitute t = 7: A(7) = 3500(1.09)^7. Step 3: Calculate (1.09)^7. First, 1.09^2 = 1.1881. Then 1.09^4 = (1.1881)^2 = 1.41158161. Then 1.09^6 = 1.41158161 * 1.1881 = 1.677100... (more precisely, 1.09^3 = 1.295029, 1.09^6 = (1.295029)^2 = 1.677100...). Finally, 1.09^7 = 1.677100... * 1.09 = 1.828039... (using precise calculation: 1.09^7 = 1.828039120...). Step 4: Multiply by 3500: 3500 * 1.828039120 = 6398.13692. Step 5: Round to the nearest dollar: 6398. The answer is 6398.

7. A population of 3,125 bacteria decreases by 20% each hour. How many bacteria remain after 3 hours? Answer: 1,600 Solution: Initial population a = 3125. Decay rate = 20% = 0.20, so decay factor b = 1 - 0.20 = 0.80. Model: P(t) = 3125 * (0.80)^t.
   Full step-by-step solution

   Step 1: Initial population a = 3125. Decay rate = 20% = 0.20, so decay factor b = 1 - 0.20 = 0.80. Model: P(t) = 3125 * (0.80)^t. Step 2: For t = 3 hours: P(3) = 3125 * (0.80)^3. Step 3: Compute (0.80)^3 = 0.80 * 0.80 * 0.80 = 0.512. Step 4: Multiply: 3125 * 0.512 = 1600. The answer is 1,600 bacteria.