Exponential Parameters

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Answer Key & Explanations

Exponential Parameters · Grade 11 · Worksheet 3

1. 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.

2. Noah is a financial analyst tracking the value of a rare vintage car collection. The total value of the collection, in thousands of dollars, is modeled by the exponential function V(t) = 280(1.09)^t, where t is the number of years since the collection was first appraised. Interpret the meaning of the parameters 280 and 1.09 in the context of this situation. Then, determine the value of the collection after 8 years, rounding your answer to the nearest thousand dollars. Answer: 558 Solution: Interpret the parameters. The function is V(t) = 280(1.09)^t. The number 280 is the initial value a.
   Full step-by-step solution

   Step 1: Interpret the parameters. The function is V(t) = 280(1.09)^t. The number 280 is the initial value a. When t = 0, V(0) = 280(1.09)^0 = 280 * 1 = 280. So 280 represents the initial value of the car collection in thousands of dollars at the time of the first appraisal, which is $280,000. The number 1.09 is the growth factor b. Since 1.09 > 1, it indicates exponential growth. Each year, the value is multiplied by 1.09, meaning it increases by 9% per year. Step 2: Find the value after 8 years. Substitute t = 8 into the function: V(8) = 280(1.09)^8. Step 3: Calculate (1.09)^8. First, (1.09)^2 = 1.09 * 1.09 = 1.1881. Then (1.09)^4 = 1.1881 * 1.1881 = 1.41158161. Then (1.09)^8 = 1.41158161 * 1.41158161 = 1.992562... (more precisely, using a calculator: 1.09^8 = 1.992562637). Step 4: Multiply by 280: V(8) = 280 * 1.992562637 = 557.91753836. Step 5: Round to the nearest thousand dollars. Since the value is in thousands of dollars, 557.9175 thousand rounds to 558 thousand dollars. The answer is 558.

3. Mere's bacterial culture grows according to P(t) = 3200(1.18)^t, where t is time in hours. What does the number 3200 represent? What does the number 1.18 represent? Answer: 3200 represents the initial population of bacteria; 1.18 represents the growth factor per hour Solution: Identify the exponential function form: P(t) = a·b^t Compare to given function: P(t) = 3200(1.18)^t Parameter a = 3200 represents the initial value when t = 0 When t = 0: P(0) = 3200(1.18)^0 = 3200(1) = 3200 Parameter b = 1.18 represents the growth factor per time unit Since b > 1, this…
   Full step-by-step solution

   Step 1: Identify the exponential function form: P(t) = a·b^t Step 2: Compare to given function: P(t) = 3200(1.18)^t Step 3: Parameter a = 3200 represents the initial value when t = 0 Step 4: When t = 0: P(0) = 3200(1.18)^0 = 3200(1) = 3200 Step 5: Parameter b = 1.18 represents the growth factor per time unit Step 6: Since b > 1, this indicates growth, and 1.18 means the population multiplies by 1.18 each hour Step 7: Therefore, 3200 is the initial population size, and 1.18 is the hourly growth factor

4. A pharmaceutical company is testing a new drug that is metabolized by the body following an exponential decay model. The concentration of the drug in the bloodstream is given by C(t) = 120 × e^(-0.15t), where C is measured in milligrams per liter and t is time in hours. Determine how many hours it will take for the drug concentration to decrease to 30 mg/L. Answer: 9.24 Solution: Set up the equation using the given function: 30 = 120 × e^(-0.15t) Divide both sides by 120: 30/120 = e^(-0.15t) Simplify: 0.25 = e^(-0.15t) Take the natural logarithm of both sides: ln(0.25) = ln(e^(-0.15t)) Simplify using logarithm properties: ln(0.25) = -0.15t Calculate ln(0.25): ln(0.25) ≈…
   Full step-by-step solution

   Step 1: Set up the equation using the given function: 30 = 120 × e^(-0.15t) Step 2: Divide both sides by 120: 30/120 = e^(-0.15t) Step 3: Simplify: 0.25 = e^(-0.15t) Step 4: Take the natural logarithm of both sides: ln(0.25) = ln(e^(-0.15t)) Step 5: Simplify using logarithm properties: ln(0.25) = -0.15t Step 6: Calculate ln(0.25): ln(0.25) ≈ -1.3863 Step 7: Substitute: -1.3863 = -0.15t Step 8: Solve for t: t = -1.3863 / -0.15 Step 9: Calculate: t ≈ 9.242 Step 10: Round to two decimal places: t ≈ 9.24 hours The answer is 9.24.

5. Aroha's investment grows according to V(t) = 375(1.15)^t, where t is in years. What does 375 represent? What does 1.15 represent? Answer: 375 represents the initial investment amount in dollars, and 1.15 represents the annual growth factor (15% growth rate) 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) = 375(1.15)^0 = 375(1) = 375. This means 375 is the initial investment amount.
   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) = 375(1.15)^0 = 375(1) = 375. This means 375 is the initial investment amount. Step 3: The base 1.15 indicates the investment grows by a factor of 1.15 each year, which represents a 15% annual growth rate. Step 4: Therefore, 375 represents the initial investment amount in dollars, and 1.15 represents the annual growth factor (15% growth rate).

6. Mason is analyzing the depreciation of a high-end manufacturing robot. The robot's value in thousands of dollars is modeled by the exponential function V(t) = 85(0.88)^t, where t is the number of years since the robot was purchased. Interpret the meaning of the parameters 85 and 0.88 in the context of this problem. Answer: 85 represents the initial value of the robot in thousands of dollars ($85,000). 0.88 represents the annual decay factor, meaning the robot retains 88% of its value each year, or depreciates by 12% per year. Solution: Identify the general form of an exponential function: f(t) = a * b^t, where a is the initial value (when t = 0) and b is the growth or decay factor per unit of time. In V(t) = 85(0.88)^t, the parameter a = 85.
   Full step-by-step solution

   Step 1: Identify the general form of an exponential function: f(t) = a * b^t, where a is the initial value (when t = 0) and b is the growth or decay factor per unit of time. Step 2: In V(t) = 85(0.88)^t, the parameter a = 85. This means when t = 0 (the time of purchase), the robot's value is 85 thousand dollars, or $85,000. Step 3: The parameter b = 0.88. Since 0.88 is between 0 and 1, this indicates exponential decay. Each year (t increases by 1), the robot's value is multiplied by 0.88, meaning it retains 88% of its value from the previous year. Step 4: To find the annual depreciation rate, subtract the decay factor from 1: 1 - 0.88 = 0.12, or 12%. So the robot loses 12% of its value each year. The answer is: 85 represents the initial value of $85,000, and 0.88 represents the annual decay factor of 88% (12% depreciation per year).

7. A radioactive substance decays exponentially, and its mass is measured over time. On a coordinate plane, the decay is represented by a curve passing through points (0, 800) and (3, 100), where the x-axis is time in hours and the y-axis is mass in grams. Determine the half-life of the substance. Answer: 1 Solution: We are told the decay is exponential, so the mass follows: m(t) = m0 * e^(k t) where m0 is initial mass, k is the decay constant (negative), and t is time in hours.
   Full step-by-step solution

   We are told the decay is exponential, so the mass follows: m(t) = m0 * e^(k t) where m0 is initial mass, k is the decay constant (negative), and t is time in hours. --- **Step 1: Find m0 and k using given points** Point (0, 800): m(0) = m0 * e^(k*0) = m0 = 800. So m(t) = 800 * e^(k t). Point (3, 100): 100 = 800 * e^(k * 3) Divide both sides by 800: 100/800 = 1/8 = e^(3k) So e^(3k) = 1/8. --- **Step 2: Solve for k** Take natural log of both sides: 3k = ln(1/8) = ln(1) - ln(8) = 0 - ln(8) = -ln(8). Thus k = -ln(8) / 3. Since 8 = 2^3, ln(8) = 3 ln(2). So k = - (3 ln(2)) / 3 = -ln(2). So k = -ln(2). --- **Step 3: Write the formula with k** m(t) = 800 * e^(-ln(2) * t) But e^(-ln(2) * t) = (e^(ln(2)))^(-t) = 2^(-t). So m(t) = 800 * 2^(-t). --- **Step 4: Find half-life** Half-life is the time T such that m(T) = m0/2 = 400. So 400 = 800 * 2^(-T) Divide by 800: 1/2 = 2^(-T) That is 2^(-1) = 2^(-T) Thus -1 = -T So T = 1. --- **Step 5: Conclusion** The half-life is 1 hour. --- **Final answer:** 1