Exponential Parameters

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Answer Key & Explanations

Exponential Parameters · Grade 11 · Worksheet 1

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. Charlotte's investment grows according to the function V(t) = 8400(1.083)^t, where t is time in years. What does the number 8400 represent? What does the number 1.083 represent? Answer: 8400 represents the initial investment amount in dollars, and 1.083 represents the annual growth factor, indicating an 8.3% annual increase. Solution: The function is in the form V(t) = a * b^t, where a is the initial value and b is the growth factor per time period. When t = 0, V(0) = 8400 * (1.083)^0 = 8400 * 1 = 8400.
   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 per time period. Step 2: When t = 0, V(0) = 8400 * (1.083)^0 = 8400 * 1 = 8400. This means at time zero (the start), the investment is worth $8400. So 8400 is the initial investment. Step 3: The base b = 1.083 is greater than 1, so it represents growth. Each year, the investment is multiplied by 1.083, meaning it grows by 8.3% per year (since 1.083 = 1 + 0.083, and 0.083 = 8.3%). Step 4: Therefore, 8400 is the initial investment amount, and 1.083 is the annual growth factor corresponding to an 8.3% annual increase.

3. Emma is a financial analyst tracking the value of a vintage watch collection. She models the total value of the collection, in thousands of dollars, using the exponential function V(t) = 85(1.12)^t, where t is the number of years since she began tracking the collection in 2020. Interpret the meaning of the parameters 85 and 1.12 in the context of the watch collection's value. Then, determine the value of the collection after 5 years, rounding your answer to the nearest thousand dollars. Answer: 150 Solution: Interpret the parameters. The function is V(t) = 85(1.12)^t. This is in the standard exponential form f(t) = a * b^t, where a is the initial value and b is the growth/decay factor.
   Full step-by-step solution

   Step 1: Interpret the parameters. The function is V(t) = 85(1.12)^t. This is in the standard exponential form f(t) = a * b^t, where a is the initial value and b is the growth/decay factor. When t = 0, V(0) = 85 * (1.12)^0 = 85 * 1 = 85. Therefore, the parameter 85 represents the initial value of the watch collection in thousands of dollars at the start of tracking in 2020, which is $85,000. The base b = 1.12. Since 1.12 > 1, the function represents exponential growth. Each year, the value is multiplied by 1.12, meaning the collection retains 100% of its previous value plus an additional 12%, so the value increases by 12% per year. Step 2: Find the value after 5 years. Substitute t = 5 into the function: V(5) = 85(1.12)^5. Step 3: Calculate (1.12)^5. 1.12^2 = 1.2544. 1.12^4 = (1.2544)^2 = 1.57351936. 1.12^5 = 1.57351936 * 1.12 = 1.7623416832. Step 4: Multiply by 85. V(5) = 85 * 1.7623416832 = 149.799043072. Step 5: Round to the nearest thousand dollars. The value is approximately 150 thousand dollars, or $150,000. The answer is 150.

4. Emma is tracking the growth of a rare fungus in a laboratory. She models the area covered by the fungus (in square centimeters) using the exponential function A(t) = 9(1.13)^t, where t is the number of weeks since the start of the experiment. Interpret the meaning of the parameters 9 and 1.13 in the context of the fungus's growth. Then, determine the area covered by the fungus after 5 weeks, rounding your answer to the nearest square centimeter. Answer: 17 Solution: Interpret the parameters. The function is A(t) = 9(1.13)^t. The parameter 9 is the initial value a, meaning at the start of the experiment (t = 0), the fungus covered 9 square centimeters.
   Full step-by-step solution

   Step 1: Interpret the parameters. The function is A(t) = 9(1.13)^t. The parameter 9 is the initial value a, meaning at the start of the experiment (t = 0), the fungus covered 9 square centimeters. The parameter 1.13 is the growth factor b, meaning each week the area is multiplied by 1.13, which represents a 13% increase per week. Step 2: Find the area after 5 weeks. Substitute t = 5 into the function: A(5) = 9(1.13)^5. Step 3: Calculate (1.13)^5. First, (1.13)^2 = 1.2769. Then (1.13)^4 = (1.2769)^2 = 1.63047361. Multiply by 1.13 to get (1.13)^5: 1.63047361 * 1.13 = 1.8424351793. Step 4: Multiply by 9: A(5) = 9 * 1.8424351793 = 16.5819166137. Step 5: Round to the nearest whole square centimeter: 16.5819 rounds to 17. The answer is 17.

5. 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.

6. A financial analyst is modeling the depreciation of a company's equipment using the exponential decay function V(t) = V₀e^(-0.12t), where V(t) is the value in thousands of dollars after t years, and V₀ is the initial value. If the equipment was originally worth $80,000, determine how many years it will take for the equipment to be worth exactly half its original value. Answer: 5.78 Solution: The initial value V₀ = 80 (in thousands of dollars) Half the original value is 80/2 = 40 (in thousands of dollars) Set up the equation: 40 = 80e^(-0.12t) Divide both sides by 80: 0.5 = e^(-0.12t) Take natural logarithm of both sides: ln(0.5) = ln(e^(-0.12t)) Simplify: ln(0.5) = -0.12t Calculate…
   Full step-by-step solution

   Step 1: The initial value V₀ = 80 (in thousands of dollars) Step 2: Half the original value is 80/2 = 40 (in thousands of dollars) Step 3: Set up the equation: 40 = 80e^(-0.12t) Step 4: Divide both sides by 80: 0.5 = e^(-0.12t) Step 5: Take natural logarithm of both sides: ln(0.5) = ln(e^(-0.12t)) Step 6: Simplify: ln(0.5) = -0.12t Step 7: Calculate ln(0.5) ≈ -0.693147 Step 8: Solve for t: t = -0.693147 / -0.12 = 5.776225 Step 9: Round to two decimal places: t ≈ 5.78 years The answer is 5.78.

7. Sophia's investment grows according to V(t) = 1600(1.06)^t, where t is time in years. What does the number 1600 represent? What does the number 1.06 represent? Answer: 1600 represents the initial investment amount in dollars, and 1.06 represents the annual growth factor, meaning the investment grows by 6% each year. Solution: Identify the exponential function form: V(t) = a·b^t Compare to given function: V(t) = 1600(1.06)^t The parameter 'a' = 1600 represents the initial value when t = 0 When t = 0: V(0) = 1600(1.06)^0 = 1600(1) = 1600 The parameter 'b' = 1.06 represents the growth factor Since b > 1, this indicates…
   Full step-by-step solution

   Step 1: Identify the exponential function form: V(t) = a·b^t Step 2: Compare to given function: V(t) = 1600(1.06)^t Step 3: The parameter 'a' = 1600 represents the initial value when t = 0 Step 4: When t = 0: V(0) = 1600(1.06)^0 = 1600(1) = 1600 Step 5: The parameter 'b' = 1.06 represents the growth factor Step 6: Since b > 1, this indicates growth Step 7: The growth rate is b - 1 = 1.06 - 1 = 0.06 = 6% Step 8: Therefore, 1600 is the initial investment and 1.06 means 6% annual growth