Exponential Parameters

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Answer Key & Explanations

Exponential Parameters · Grade 11 · Worksheet 2

1. Aroha's investment grows according to V(t) = 950(1.12)^t, where V is the value in dollars and t is time in years. What does the number 950 represent? What does the number 1.12 represent? Answer: 950 represents the initial investment amount of $950, and 1.12 represents the annual growth factor where the investment increases by 12% each year Solution: The function is in the form V(t) = a·b^t, where a is the initial value and b is the growth factor. When t = 0, V(0) = 950(1.12)^0 = 950(1) = 950. This means the investment starts at $950.
   Full step-by-step solution

   Step 1: The function is in the form V(t) = a·b^t, where a is the initial value and b is the growth factor. Step 2: When t = 0, V(0) = 950(1.12)^0 = 950(1) = 950. This means the investment starts at $950. Step 3: The growth factor b = 1.12 means the investment multiplies by 1.12 each year, which represents a 12% annual increase (since 1.12 = 1 + 0.12). Step 4: Therefore, 950 represents the initial investment amount, and 1.12 represents the annual growth factor indicating a 12% yearly increase.

2. Matiu is studying the depreciation of a specialized piece of forestry equipment. He models its value using the exponential function V(t) = 48000 × (0.78)^t, where V(t) is the value in dollars after t years. Interpret the meaning of the parameters 48000 and 0.78 in the context of the equipment's value over time. Answer: 48000 is the initial value of the equipment in dollars when t = 0; 0.78 is the decay factor, meaning the equipment retains 78% of its value each year. Solution: Identify the general form V(t) = a * b^t. Here, a = 48000 and b = 0.78. When t = 0, V(0) = 48000 * (0.78)^0 = 48000 * 1 = 48000.
   Full step-by-step solution

   Step 1: Identify the general form V(t) = a * b^t. Here, a = 48000 and b = 0.78. Step 2: When t = 0, V(0) = 48000 * (0.78)^0 = 48000 * 1 = 48000. So 48000 represents the initial value of the equipment when it was first purchased (t = 0 years). Step 3: The base b = 0.78. Since 0.78 is less than 1, the function models exponential decay. Each year, the value is multiplied by 0.78, meaning the equipment retains 78% of its value from the previous year (or loses 22% of its value each year). Step 4: Therefore, 48000 means the equipment was initially worth $48,000, and 0.78 means the equipment depreciates by 22% annually, retaining 78% of its value each year.

3. A research team 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 log of both sides: 3k = ln(2) k = ln(2) / 3 We can keep it as an expression for now. --- Step 4: Write the general formula for P(t). P(t) = 500 * e^( (ln(2)/3) * t ) --- Step 5: Find P(9). t = 9: P(9) = 500 * e^( (ln(2)/3) * 9 ) Simplify exponent: (ln(2)/3) * 9 = 3 * ln(2) = ln(2^3) = ln(8) So: P(9) = 500 * e^(ln(8)) Since e^(ln(8)) = 8: P(9) = 500 * 8 = 4000 --- Step 6: Conclusion. The continuous growth rate constant k = ln(2)/3, and the population after 9 hours is 4000 bacteria.

4. Aroha is studying the population decline of a native bird species on an island due to habitat loss. She models the population using the exponential decay function P(t) = 1200(0.88)^t, where P(t) is the number of birds remaining after t years. Interpret the meaning of the parameters 1200 and 0.88 in the context of the bird population. Then, determine how many years it will take for the population to decline to 600 birds, rounding your answer to the nearest tenth of a year. Answer: 5.4 Solution: Interpret the parameters. The value 1200 is the initial population of birds at t = 0 years.
   Full step-by-step solution

   Step 1: Interpret the parameters. The value 1200 is the initial population of birds at t = 0 years. The value 0.88 is the decay factor; each year, the population is multiplied by 0.88, meaning it retains 88% of the previous year's population, so it decreases by 12% per year. Step 2: Set up the equation for when the population is 600 birds: 1200(0.88)^t = 600. Step 3: Divide both sides by 1200: (0.88)^t = 600/1200 = 0.5. Step 4: Take the natural logarithm of both sides: ln((0.88)^t) = ln(0.5). Step 5: Use the power rule of logarithms: t * ln(0.88) = ln(0.5). Step 6: Solve for t: t = ln(0.5) / ln(0.88). Step 7: Calculate: ln(0.5) ≈ -0.693147, ln(0.88) ≈ -0.127833. Step 8: t = (-0.693147) / (-0.127833) ≈ 5.422. Step 9: Round to the nearest tenth: t ≈ 5.4 years. The answer is 5.4.

5. Liam is studying the spread of information through social networks. He models the number of people who have seen a viral post using the function P(t) = 500 × 2^(0.3t), where t is the number of hours since the post was first shared. If Liam wants to know when exactly 4,000 people will have seen the post, what equation should he solve and what is the solution for t? Answer: t = 10 Solution: P(t) = 500 × 2^(0.3t) We want to find t when P(t) = 4000. Set up the equation. 4000 = 500 × 2^(0.3t) Divide both sides by 500 to isolate the exponential term.
   Full step-by-step solution

   We are given the function: P(t) = 500 × 2^(0.3t) We want to find t when P(t) = 4000. Step 1: Set up the equation. 4000 = 500 × 2^(0.3t) Step 2: Divide both sides by 500 to isolate the exponential term. 4000 / 500 = 2^(0.3t) 8 = 2^(0.3t) Step 3: Recognize that 8 is a power of 2. 8 = 2^3 So we have: 2^3 = 2^(0.3t) Step 4: Since the bases are the same (base 2), we can equate the exponents. 3 = 0.3t Step 5: Solve for t. t = 3 / 0.3 t = 30 / 3 t = 10 Step 6: Conclusion. The time when 4000 people have seen the post is t = 10 hours.

6. Noah is tracking the growth of a rare plant species. The height of the plant, in centimeters, after t days is modeled by the exponential function H(t) = 6(1.21)^t. Interpret the meaning of the parameters 6 and 1.21 in the context of the plant's growth. Answer: The parameter 6 represents the initial height of the plant (6 cm at day 0), and 1.21 represents the daily growth factor, meaning the plant's height increases by 21% each day. Solution: Identify the general form of an exponential function: H(t) = a * b^t, where a is the initial value (when t = 0) and b is the growth factor per unit time.
   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 (when t = 0) and b is the growth factor per unit time. Step 2: In the given function H(t) = 6(1.21)^t, substitute t = 0: H(0) = 6 * (1.21)^0 = 6 * 1 = 6. This means the plant's height at day 0 is 6 cm. So, the parameter 6 represents the initial height. Step 3: The base 1.21 is the factor by which the height multiplies each day. For example, at t = 1: H(1) = 6 * 1.21 = 7.26 cm. At t = 2: H(2) = 6 * (1.21)^2 = 6 * 1.4641 = 8.7846 cm. The ratio H(2)/H(1) = 8.7846 / 7.26 = 1.21, confirming the daily multiplication factor is 1.21. Step 4: A growth factor of 1.21 means the height increases by 1.21 - 1 = 0.21, or 21% each day. Final answer: The parameter 6 is the initial height (6 cm), and 1.21 is the daily growth factor, indicating a 21% increase per day.

7. Olivia is studying the growth of a bamboo plant in her backyard. She models the height of the plant using the exponential function H(t) = 15(1.4)^t, where H(t) is the height in centimeters after t weeks. Interpret the meaning of the parameters 15 and 1.4 in this context. Answer: 15 represents the initial height of the bamboo plant in centimeters at time t = 0 weeks, and 1.4 represents the weekly growth factor, meaning the plant's height increases by 40% each week. Solution: Identify the general form of an exponential function: H(t) = a * b^t, where a is the initial value (when t = 0) and b is the growth factor per unit time.
   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 (when t = 0) and b is the growth factor per unit time. Step 2: In H(t) = 15(1.4)^t, when t = 0, H(0) = 15 * (1.4)^0 = 15 * 1 = 15. So the parameter 15 represents the initial height of the bamboo plant at the start of observation, which is 15 centimeters. Step 3: The base 1.4 is the growth factor. Since 1.4 > 1, it indicates growth. The growth rate is (1.4 - 1) * 100% = 0.4 * 100% = 40% per week. This means each week, the plant's height is multiplied by 1.4, so it grows by 40% of its previous height each week. Step 4: Therefore, 15 is the initial height in centimeters, and 1.4 is the weekly growth factor representing a 40% increase per week.