Exponential Growth and Decay

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Answer Key & Explanations

Exponential Growth and Decay · Grade 10 · Worksheet 1

1. A population of bacteria starts at 450 and triples every 4 hours. Write an exponential function P(t) to model the population after t hours. Then, determine the population after 12 hours. Answer: 12150 Solution: Identify the initial population a = 450. The population triples every 4 hours, so the growth factor per 4 hours is 3. The hourly growth factor b is the 4th root of 3: b = 3^(1/4).
   Full step-by-step solution

   Step 1: Identify the initial population a = 450. Step 2: The population triples every 4 hours, so the growth factor per 4 hours is 3. The hourly growth factor b is the 4th root of 3: b = 3^(1/4). Step 3: The exponential model is P(t) = 450 * (3^(1/4))^t = 450 * 3^(t/4). Step 4: To find the population after 12 hours, substitute t = 12: P(12) = 450 * 3^(12/4) = 450 * 3^3. Step 5: Calculate 3^3 = 27. Step 6: Multiply: 450 * 27 = 12150. The population after 12 hours is 12150.

2. Hana is monitoring the decay of a radioactive isotope in a laboratory. A graph shows the mass of the isotope over time, with the mass decreasing exponentially. At time t = 0 hours, the mass is 240 grams. After 16 hours, the mass has decayed to 60 grams. The graph passes through the points (0, 240) and (16, 60). What is the half-life of the isotope in hours? Answer: 8 hours Solution: Write the exponential decay model: A(t) = A_0 * b^t, where A_0 = 240 grams and b is the hourly decay factor. At t = 16 hours, A(16) = 60. So 60 = 240 * b^16.
   Full step-by-step solution

   Step 1: Write the exponential decay model: A(t) = A_0 * b^t, where A_0 = 240 grams and b is the hourly decay factor. At t = 16 hours, A(16) = 60. So 60 = 240 * b^16. Step 2: Divide both sides by 240: 60/240 = b^16 => 1/4 = b^16. Step 3: Take the 16th root: b = (1/4)^(1/16). Alternatively, note that 1/4 = (1/2)^2, so b^16 = (1/2)^2, so b = (1/2)^(2/16) = (1/2)^(1/8). Step 4: The half-life is the time T such that A(T) = 240 * b^T = 120 (half of 240). So 240 * b^T = 120 => b^T = 1/2. Step 5: Substitute b = (1/2)^(1/8): ((1/2)^(1/8))^T = 1/2 => (1/2)^(T/8) = (1/2)^1 => T/8 = 1 => T = 8. The half-life is 8 hours.

3. Aroha is observing a rare plant species in a protected forest. The population of the plants, P(t), is modeled by the exponential decay function P(t) = 8500(0.91)^t, where t is the number of years since the study began. The graph of this function shows a steep decline that gradually levels off. What is the annual percentage rate of decay of the plant population? Answer: 9% Solution: Identify the decay factor from the given model P(t) = 8500(0.91)^t. The decay factor is 0.91. Since this is exponential decay, the decay rate r is found by subtracting the decay factor from 1: r = 1 - 0.91 = 0.09.
   Full step-by-step solution

   Step 1: Identify the decay factor from the given model P(t) = 8500(0.91)^t. The decay factor is 0.91. Step 2: Since this is exponential decay, the decay rate r is found by subtracting the decay factor from 1: r = 1 - 0.91 = 0.09. Step 3: Convert the decimal rate to a percentage: 0.09 * 100 = 9%. Step 4: Therefore, the plant population decreases by 9% each year. The answer is 9%.

4. Mason is a medical researcher studying the decay of a new radioactive isotope used in cancer treatment. The isotope has a half-life of 14 days. He starts with a 240 mg sample. Using the exponential decay formula A(t) = A₀ × (1/2)^(t/h), where A₀ is the initial amount in mg, t is time in days, and h is the half-life in days, how many milligrams of the isotope remain after 42 days? Answer: 30 Solution: Identify the given values. Initial amount A₀ = 240 mg Half-life h = 14 days Time t = 42 days Determine the number of half-lives that have passed.
   Full step-by-step solution

   Step 1: Identify the given values. Initial amount A₀ = 240 mg Half-life h = 14 days Time t = 42 days Step 2: Determine the number of half-lives that have passed. Number of half-lives = t / h = 42 / 14 = 3 Step 3: Apply the exponential decay formula. A(t) = A₀ × (1/2)^(t/h) A(42) = 240 × (1/2)^(42/14) A(42) = 240 × (1/2)^3 Step 4: Calculate (1/2)^3. (1/2)^3 = 1/8 Step 5: Multiply by the initial amount. A(42) = 240 × (1/8) A(42) = 240 / 8 A(42) = 30 The answer is 30 mg.

5. Mere is monitoring the population of a rare bird species in a protected sanctuary. The initial population is 240 birds, and the population grows at a rate of 4% 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 the time in years, determine the bird population after 6 years. Round your answer to the nearest whole number. Answer: 304 Solution: Identify the given values. Initial population P₀ = 240 birds Annual growth rate r = 4% = 0.04 Time t = 6 years Write the exponential growth formula. P(t) = P₀ × (1 + r)^t Substitute the values.
   Full step-by-step solution

   Step 1: Identify the given values. Initial population P₀ = 240 birds Annual growth rate r = 4% = 0.04 Time t = 6 years Step 2: Write the exponential growth formula. P(t) = P₀ × (1 + r)^t Step 3: Substitute the values. P(6) = 240 × (1 + 0.04)^6 P(6) = 240 × (1.04)^6 Step 4: Calculate (1.04)^6. 1.04^2 = 1.0816 1.04^4 = (1.0816)^2 = 1.16985856 1.04^6 = 1.04^4 × 1.04^2 = 1.16985856 × 1.0816 = 1.265319... (approximately 1.265319) Step 5: Multiply by the initial population. P(6) = 240 × 1.265319 P(6) = 303.67656 Step 6: Round to the nearest whole number. 303.67656 rounds to 304. The answer is 304 birds.

6. Aroha is studying the population growth of a rare bird species on a protected island. The initial population is 125 birds, and the population grows at a constant annual 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 expressed as a decimal, and t is the number of years, determine the bird population after 15 years. Round your answer to the nearest whole number. Answer: 345 Solution: Identify the given values. Initial population P₀ = 125 birds Annual growth rate r = 7% = 0.07 Time t = 15 years Write the exponential growth formula. P(t) = P₀(1 + r)^t Substitute the values into the formula.
   Full step-by-step solution

   Step 1: Identify the given values. Initial population P₀ = 125 birds Annual growth rate r = 7% = 0.07 Time t = 15 years Step 2: Write the exponential growth formula. P(t) = P₀(1 + r)^t Step 3: Substitute the values into the formula. P(15) = 125(1 + 0.07)^15 P(15) = 125(1.07)^15 Step 4: Calculate (1.07)^15. 1.07^2 = 1.1449 1.07^4 = (1.1449)^2 = 1.31079601 1.07^8 = (1.31079601)^2 ≈ 1.71818618 1.07^12 = 1.07^8 × 1.07^4 ≈ 1.71818618 × 1.31079601 ≈ 2.25219159 1.07^15 = 1.07^12 × 1.07^3 First, 1.07^3 = 1.07^2 × 1.07 = 1.1449 × 1.07 = 1.225043 Then, 1.07^15 ≈ 2.25219159 × 1.225043 ≈ 2.759031 Step 5: Multiply by the initial population. P(15) = 125 × 2.759031 = 344.878875 Step 6: Round to the nearest whole number. 344.878875 rounds to 345. The answer is 345 birds.