Exponential Growth and Decay

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Answer Key & Explanations

Exponential Growth and Decay · Grade 10 · Worksheet 3

1. A rare species of orchid is being cultivated in a botanical garden. The number of orchids follows the exponential growth model N(t) = 80 × 2^(t/3), where t is the time in years. How many orchids will there be after 9 years? Answer: 640 Solution: The exponential growth model is N(t) = 80 × 2^(t/3) Substitute t = 9 into the function: N(9) = 80 × 2^(9/3) Simplify the exponent: 9/3 = 3, so N(9) = 80 × 2^3 Calculate 2^3 = 8 Multiply: 80 × 8 = 640 Therefore, there will be 640 orchids after 9 years.
   Full step-by-step solution

   Step 1: The exponential growth model is N(t) = 80 × 2^(t/3) Step 2: Substitute t = 9 into the function: N(9) = 80 × 2^(9/3) Step 3: Simplify the exponent: 9/3 = 3, so N(9) = 80 × 2^3 Step 4: Calculate 2^3 = 8 Step 5: Multiply: 80 × 8 = 640 Step 6: Therefore, there will be 640 orchids after 9 years. The answer is 640.

2. Mason invests $15,000 in a savings account that earns 8.4% annual interest compounded monthly. Write the exponential function A(t) that models the amount in the account after t years, and then determine the amount after 9 years, rounded to the nearest dollar. Answer: A(t) = 15000(1.007)^(12t); $32,178 Solution: The initial principal P = $15,000. The annual interest rate r = 8.4% = 0.084. Compounded monthly means n = 12 times per year.
   Full step-by-step solution

   Step 1: The initial principal P = $15,000. The annual interest rate r = 8.4% = 0.084. Compounded monthly means n = 12 times per year. Step 2: The monthly interest rate is r/n = 0.084/12 = 0.007. Step 3: The exponential model is A(t) = P(1 + r/n)^(nt) = 15000(1 + 0.007)^(12t) = 15000(1.007)^(12t). Step 4: For t = 9 years, compute A(9) = 15000(1.007)^(12*9) = 15000(1.007)^108. Step 5: Calculate (1.007)^108. Using a calculator: (1.007)^108 ≈ 2.1452. Step 6: Multiply: 15000 × 2.1452 = 32,178. Step 7: Rounded to the nearest dollar, the amount after 9 years is $32,178. Final answer: A(t) = 15000(1.007)^(12t); $32,178

3. Aroha is monitoring the growth of a rare fern species in a conservation reserve. The initial population of ferns is 135, and the population grows at a rate of 7% per year. Using the exponential growth model P(t) = P₀(1 + r)^t, where P₀ is the initial population, r is the annual growth rate as a decimal, and t is time in years, determine the population of ferns after 9 years. Round your answer to the nearest whole number. Answer: 248 Solution: Identify the given values. Initial population P₀ = 135 Annual growth rate r = 7% = 0.07 Time t = 9 years Write the exponential growth formula. P(t) = P₀(1 + r)^t P(9) = 135(1 + 0.07)^9 Calculate the growth factor.
   Full step-by-step solution

   Step 1: Identify the given values. Initial population P₀ = 135 Annual growth rate r = 7% = 0.07 Time t = 9 years Step 2: Write the exponential growth formula. P(t) = P₀(1 + r)^t P(9) = 135(1 + 0.07)^9 Step 3: Calculate the growth factor. 1 + 0.07 = 1.07 Step 4: Compute (1.07)^9. 1.07^2 = 1.1449 1.07^4 = (1.1449)^2 = 1.31079601 1.07^8 = (1.31079601)^2 = 1.71818618 1.07^9 = 1.71818618 * 1.07 = 1.83845921 Step 5: Multiply by the initial population. P(9) = 135 * 1.83845921 P(9) = 248.19199335 Step 6: Round to the nearest whole number. 248.19199335 rounds to 248. The answer is 248.

4. Isabella is studying a population of bacteria that doubles every 9 hours. She starts with 12 bacteria cells. She creates a graph where the x-axis represents time in hours (t) and the y-axis represents the number of bacteria (N). The graph shows the exponential curve N(t) = 12 * 2^(t/9). If Isabella draws a horizontal line at N = 96, at what time (in hours) will the population curve intersect this horizontal line? Round your answer to the nearest whole hour. Answer: 27 Solution: Set the equation equal to the target population. 12 * 2^(t/9) = 96 Divide both sides by 12. 2^(t/9) = 8 Rewrite 8 as a power of 2.
   Full step-by-step solution

   Step 1: Set the equation equal to the target population. 12 * 2^(t/9) = 96 Step 2: Divide both sides by 12. 2^(t/9) = 8 Step 3: Rewrite 8 as a power of 2. 8 = 2^3 Step 4: Since the bases are equal, set the exponents equal to each other. t/9 = 3 Step 5: Multiply both sides by 9 to solve for t. t = 27 The answer is 27 hours.

5. Olivia is tracking the population of a rare species of orchid in a protected forest reserve. The current population is 1,200 orchids, and the population is decreasing at a rate of 15% per year due to habitat loss. Write an exponential decay model to represent the orchid population after t years, and use it to determine the population after 10 years, rounding to the nearest whole orchid. Answer: 236 Solution: Identify the initial population and decay rate. Initial population P0 = 1,200. Decay rate = 15% per year = 0.15.
   Full step-by-step solution

   Step 1: Identify the initial population and decay rate. Initial population P0 = 1,200. Decay rate = 15% per year = 0.15. Step 2: Find the decay factor. Since the population decreases by 15%, the remaining percentage each year is 100% - 15% = 85% = 0.85. Step 3: Write the exponential decay model. P(t) = P0 * (decay factor)^t = 1,200 * (0.85)^t. Step 4: Substitute t = 10 into the model. P(10) = 1,200 * (0.85)^10. Step 5: Calculate (0.85)^10. 0.85^2 = 0.7225 0.85^4 = (0.7225)^2 = 0.52200625 0.85^5 = 0.52200625 * 0.85 = 0.4437053125 0.85^10 = (0.85^5)^2 = (0.4437053125)^2 = 0.196874... (more precisely, 0.85^10 = 0.196874404...) Step 6: Multiply by 1,200. P(10) = 1,200 * 0.196874404 = 236.2492848. Step 7: Round to the nearest whole orchid. 236.2492848 rounds to 236. The answer is 236 orchids.

6. Noah is a microbiologist studying a bacterial culture. The initial population is 1,600 bacteria, and the population triples every 6 hours. Using the exponential growth model P(t) = P₀ × 3^(t/k), where P₀ is the initial population, t is time in hours, and k is the tripling time, determine the population after 18 hours. Answer: 43200 Solution: Identify the given values. Initial population P₀ = 1,600 bacteria Tripling time k = 6 hours Time t = 18 hours Calculate the number of tripling periods in 18 hours.
   Full step-by-step solution

   Step 1: Identify the given values. Initial population P₀ = 1,600 bacteria Tripling time k = 6 hours Time t = 18 hours Step 2: Calculate the number of tripling periods in 18 hours. Number of tripling periods = t / k = 18 / 6 = 3 Step 3: Apply the exponential growth formula. P(t) = P₀ × 3^(t/k) P(18) = 1,600 × 3^(18/6) P(18) = 1,600 × 3^3 Step 4: Calculate 3^3. 3^3 = 3 × 3 × 3 = 27 Step 5: Multiply to find the final population. P(18) = 1,600 × 27 = 43,200 The answer is 43,200 bacteria.

7. A sample of a radioactive isotope has an initial mass of 240 grams. Its half-life is 8 years. Write an exponential decay model A(t) for the mass remaining after t years, and use it to find the mass remaining after 24 years. Answer: 30 grams Solution: The general exponential decay model is A(t) = A₀ * (1/2)^(t / h), where A₀ is the initial amount, h is the half-life, and t is time. Here, A₀ = 240 grams and h = 8 years.
   Full step-by-step solution

   Step 1: The general exponential decay model is A(t) = A₀ * (1/2)^(t / h), where A₀ is the initial amount, h is the half-life, and t is time. Here, A₀ = 240 grams and h = 8 years. So the model is A(t) = 240 * (1/2)^(t/8). Step 2: To find the mass after 24 years, substitute t = 24 into the model: A(24) = 240 * (1/2)^(24/8). Step 3: Simplify the exponent: 24/8 = 3. So A(24) = 240 * (1/2)^3. Step 4: Calculate (1/2)^3 = 1/8. Step 5: Multiply: 240 * (1/8) = 240/8 = 30. Step 6: The mass remaining after 24 years is 30 grams.