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Exponential Growth and Decay

Grade 10 · Mathematics · Worksheet 1

  1. Emma is studying the population growth of a rare species of orchid in a protected forest reserve. The current population is 125 orchids, and the population is expected to grow 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 growth rate as a decimal, and t is time in years, determine the orchid population after 9 years. Round your answer to the nearest whole number. Answer: ______________
  2. 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: ______________
  3. A population of 800 bacteria grows at a rate of 12% per hour. Write an exponential function P(t) to model the population after t hours. Then, determine the population after 5 hours, rounded to the nearest whole number. Answer: ______________
  4. Emma is studying the growth of a rare fern species in a controlled greenhouse. The initial population of ferns is 15, and the population triples every 7 months. Using the exponential growth model P(t) = P₀ × 3^(t/k), where P₀ is the initial population, t is time in months, and k is the tripling time, determine the fern population after 21 months. Answer: ______________
  5. Ava is studying the growth of a certain strain of bacteria for her biology project. She starts with a culture containing 16 bacteria. The bacteria population triples every 4 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, how many bacteria will be present after 16 hours? Answer: ______________
  6. Tane is a conservation biologist studying a population of rare native birds on a remote island. The initial population is 375 birds. The population is decreasing at a rate of 7% per year due to habitat loss. Write an exponential decay function P(t) to model the population after t years, and then use it to determine the number of birds remaining after 9 years. Round your answer to the nearest whole bird. Answer: ______________
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Answer Key & Explanations

Exponential Growth and Decay · Grade 10 · Worksheet 1

  1. Emma is studying the population growth of a rare species of orchid in a protected forest reserve. The current population is 125 orchids, and the population is expected to grow 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 growth rate as a decimal, and t is time in years, determine the orchid population after 9 years. Round your answer to the nearest whole number. Answer: 230 Solution: Identify the given values. Initial population P₀ = 125 orchids Growth rate r = 7% = 0.07 (as a decimal) Time t = 9 years Write the exponential growth model.
    Full step-by-step solution

    Step 1: Identify the given values. Initial population P₀ = 125 orchids Growth rate r = 7% = 0.07 (as a decimal) Time t = 9 years Step 2: Write the exponential growth model. P(t) = P₀ × (1 + r)^t P(9) = 125 × (1 + 0.07)^9 Step 3: Calculate the base. 1 + 0.07 = 1.07 Step 4: Raise the base to the power of 9. 1.07^9 = 1.07 × 1.07 × 1.07 × 1.07 × 1.07 × 1.07 × 1.07 × 1.07 × 1.07 Using a calculator: 1.07^9 ≈ 1.838459 Step 5: Multiply by the initial population. P(9) = 125 × 1.838459 P(9) ≈ 229.8074 Step 6: Round to the nearest whole number. 229.8074 rounds to 230. The answer is 230 orchids.

  2. 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%.

  3. A population of 800 bacteria grows at a rate of 12% per hour. Write an exponential function P(t) to model the population after t hours. Then, determine the population after 5 hours, rounded to the nearest whole number. Answer: 1410 Solution: The initial population is a = 800. The growth rate is 12% per hour, which as a decimal is 0.12. The growth factor is b = 1 + 0.12 = 1.12.
    Full step-by-step solution

    Step 1: The initial population is a = 800. The growth rate is 12% per hour, which as a decimal is 0.12. The growth factor is b = 1 + 0.12 = 1.12. Step 2: The exponential model is P(t) = 800 * (1.12)^t. Step 3: To find the population after 5 hours, substitute t = 5: P(5) = 800 * (1.12)^5. Step 4: Calculate (1.12)^5 = 1.12 * 1.12 * 1.12 * 1.12 * 1.12 = 1.7623416832. Step 5: Multiply by 800: 800 * 1.7623416832 = 1409.87334656. Step 6: Round to the nearest whole number: 1410. The answer is 1410.

  4. Emma is studying the growth of a rare fern species in a controlled greenhouse. The initial population of ferns is 15, and the population triples every 7 months. Using the exponential growth model P(t) = P₀ × 3^(t/k), where P₀ is the initial population, t is time in months, and k is the tripling time, determine the fern population after 21 months. Answer: 405 Solution: Identify the given values. Initial population P₀ = 15 ferns Tripling time k = 7 months Time t = 21 months Determine how many tripling periods occur in 21 months.
    Full step-by-step solution

    Step 1: Identify the given values. Initial population P₀ = 15 ferns Tripling time k = 7 months Time t = 21 months Step 2: Determine how many tripling periods occur in 21 months. Number of tripling periods = t/k = 21 / 7 = 3 Step 3: Apply the exponential growth formula. P(t) = P₀ × 3^(t/k) P(21) = 15 × 3^(21/7) P(21) = 15 × 3^3 Step 4: Calculate 3^3. 3^3 = 3 × 3 × 3 = 27 Step 5: Multiply by the initial population. P(21) = 15 × 27 = 405 The fern population after 21 months is 405.

  5. Ava is studying the growth of a certain strain of bacteria for her biology project. She starts with a culture containing 16 bacteria. The bacteria population triples every 4 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, how many bacteria will be present after 16 hours? Answer: 1296 Solution: Identify the given values. Initial population P₀ = 16 bacteria. Tripling time k = 4 hours.
    Full step-by-step solution

    Step 1: Identify the given values. Initial population P₀ = 16 bacteria. Tripling time k = 4 hours. Total time t = 16 hours. Step 2: Determine the number of tripling periods in the given time. Number of tripling periods = t/k = 16/4 = 4. Step 3: Apply the exponential growth formula. P(t) = P₀ × 3^(t/k) = 16 × 3^(16/4) = 16 × 3^4. Step 4: Calculate 3^4 = 3 × 3 × 3 × 3 = 81. Step 5: Multiply by the initial population. P(16) = 16 × 81 = 1296. The answer is 1296 bacteria.

  6. Tane is a conservation biologist studying a population of rare native birds on a remote island. The initial population is 375 birds. The population is decreasing at a rate of 7% per year due to habitat loss. Write an exponential decay function P(t) to model the population after t years, and then use it to determine the number of birds remaining after 9 years. Round your answer to the nearest whole bird. Answer: 195 Solution: Identify the initial population and decay rate. Initial population a = 375 birds Decay rate = 7% = 0.07 per year Determine the decay factor b.
    Full step-by-step solution

    Step 1: Identify the initial population and decay rate. Initial population a = 375 birds Decay rate = 7% = 0.07 per year Step 2: Determine the decay factor b. Since the population decreases by 7% each year, it retains 100% - 7% = 93% of its value. b = 1 - 0.07 = 0.93 Step 3: Write the exponential decay function. P(t) = 375 * (0.93)^t Step 4: Substitute t = 9 years into the function. P(9) = 375 * (0.93)^9 Step 5: Calculate (0.93)^9. 0.93^2 = 0.8649 0.93^4 = 0.8649^2 = 0.74805201 0.93^8 = 0.74805201^2 = 0.559581 0.93^9 = 0.559581 * 0.93 = 0.520410 Step 6: Multiply by the initial population. P(9) = 375 * 0.520410 = 195.15375 Step 7: Round to the nearest whole bird. 195.15375 rounds to 195 birds. The answer is 195.