Exponential Parameters

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Answer Key & Explanations

Exponential Parameters · Grade 11 · Worksheet 3

1. Kaia is studying the growth of a fern species in a forest regeneration project. The height of the fern is modeled by the exponential function H(t) = 7 × (1.21)^t, where H(t) is the height in centimeters after t months. Interpret the meaning of the parameters 7 and 1.21 in this context. Answer: The value 7 represents the initial height of the fern in centimeters when t = 0 months. The value 1.21 represents the monthly growth factor, meaning the fern's height increases by 21% each month. 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 or decay factor. In H(t) = 7 × (1.21)^t, the parameter a = 7.
   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 or decay factor. Step 2: In H(t) = 7 × (1.21)^t, the parameter a = 7. When t = 0, H(0) = 7 × (1.21)^0 = 7 × 1 = 7. This means the fern's initial height at the start of observation is 7 centimeters. Step 3: The parameter b = 1.21. Since b > 1, this represents exponential growth. The growth factor 1.21 means that each month, the fern's height is multiplied by 1.21. The percentage increase is (1.21 - 1) × 100% = 0.21 × 100% = 21% per month. Final answer: The 7 represents the initial height of 7 centimeters, and the 1.21 represents a monthly growth factor of 21% increase per month.

2. Sophia is studying a bacterial culture that grows according to the function P(t) = 750(1.12)^t, where t is time in hours. What does the number 750 represent in this context? What does the number 1.12 represent? Answer: 750 represents the initial population size of the bacterial culture, and 1.12 represents the growth factor per hour, indicating a 12% hourly growth rate Solution: Identify the exponential function format: P(t) = a·b^t Compare to given function: P(t) = 750(1.12)^t The parameter 'a' (750) represents the initial value when t = 0 When t = 0, P(0) = 750(1.12)^0 = 750(1) = 750 So 750 represents the starting population of bacteria The parameter 'b' (1.12)…
   Full step-by-step solution

   Step 1: Identify the exponential function format: P(t) = a·b^t Step 2: Compare to given function: P(t) = 750(1.12)^t Step 3: The parameter 'a' (750) represents the initial value when t = 0 Step 4: When t = 0, P(0) = 750(1.12)^0 = 750(1) = 750 Step 5: So 750 represents the starting population of bacteria Step 6: The parameter 'b' (1.12) represents the growth factor Step 7: Since b > 1, this is exponential growth Step 8: The growth rate is b - 1 = 1.12 - 1 = 0.12, or 12% per hour Step 9: So 1.12 means the population multiplies by 1.12 each hour Final answer: 750 is the initial population, 1.12 is the hourly growth factor indicating 12% growth per hour

3. 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).

4. Aroha's investment grows according to V(t) = 575(1.07)^t. What does the value 575 represent? What does the value 1.07 represent? Answer: 575 is the initial investment amount, 1.07 is the growth factor per time period Solution: The function V(t) = 575(1.07)^t represents exponential growth. The parameter 'a' (575) is the initial value when t = 0. When t = 0, V(0) = 575(1.07)^0 = 575(1) = 575.
   Full step-by-step solution

   Step 1: The function V(t) = 575(1.07)^t represents exponential growth. Step 2: The parameter 'a' (575) is the initial value when t = 0. When t = 0, V(0) = 575(1.07)^0 = 575(1) = 575. This represents the starting investment amount. Step 3: The parameter 'b' (1.07) is the growth factor. Since 1.07 > 1, this represents growth. The value 1.07 means the investment grows by 7% each time period (1.07 = 1 + 0.07). Step 4: Therefore, 575 represents the initial investment amount, and 1.07 represents the growth factor per time period.

5. Hana is studying the population decline of a rare bird species on an island. The population is modeled by the exponential decay function P(t) = 2400(0.88)^t, where P(t) is the number of birds remaining after t years. Interpret the meaning of the parameters 2400 and 0.88 in this context. Answer: 2400 is the initial population of birds, and 0.88 is the annual decay factor, meaning the population retains 88% of its size each year. Solution: The function is P(t) = 2400(0.88)^t. Compare to the standard form f(t) = a * b^t. Step 2: The parameter a is the initial value, the value when t = 0.
   Full step-by-step solution

   Step 1: The function is P(t) = 2400(0.88)^t. Compare to the standard form f(t) = a * b^t. Step 2: The parameter a is the initial value, the value when t = 0. P(0) = 2400(0.88)^0 = 2400 * 1 = 2400. So 2400 represents the initial population of birds on the island. Step 3: The parameter b is the growth or decay factor. Since 0.88 is less than 1, it represents decay. Each year, the population is multiplied by 0.88, meaning it retains 88% of the previous year's population (a 12% annual decrease). Step 4: Therefore, 2400 is the starting number of birds, and 0.88 is the annual decay factor.

6. Sophia is studying a radioactive substance that decays according to the function M(t) = 160(0.86)^t, where M is the mass in grams and t is time in years. What does the number 160 represent? What does the number 0.86 represent? Answer: 160 represents the initial mass of the substance in grams. 0.86 represents the decay factor per year, meaning the mass decreases by 14% each year. Solution: The function is M(t) = 160(0.86)^t, which follows the exponential form f(t) = a·b^t. When t = 0 (initial time), M(0) = 160(0.86)^0 = 160(1) = 160 grams. So 160 represents the initial mass.
   Full step-by-step solution

   Step 1: The function is M(t) = 160(0.86)^t, which follows the exponential form f(t) = a·b^t. Step 2: When t = 0 (initial time), M(0) = 160(0.86)^0 = 160(1) = 160 grams. So 160 represents the initial mass. Step 3: The base 0.86 is the decay factor. Since 0.86 < 1, this represents exponential decay. Step 4: The decay rate per year is 1 - 0.86 = 0.14, or 14% decrease per year. Step 5: Therefore, 160 is the initial mass in grams, and 0.86 is the decay factor showing the substance retains 86% of its mass each year.

7. Hana's investment grows according to the function V(t) = 2400(1.06)^t, where t is time in years. What does the number 2400 represent? What does the number 1.06 represent? Answer: 2400 is the initial investment amount; 1.06 is the annual growth factor Solution: The function is V(t) = 2400(1.06)^t, which follows the exponential form f(t) = a·b^t. When t = 0, V(0) = 2400(1.06)^0 = 2400(1) = 2400. This represents the starting value of the investment.
   Full step-by-step solution

   Step 1: The function is V(t) = 2400(1.06)^t, which follows the exponential form f(t) = a·b^t. Step 2: When t = 0, V(0) = 2400(1.06)^0 = 2400(1) = 2400. This represents the starting value of the investment. Step 3: The base 1.06 indicates the growth factor per time period. Since t is in years, this means the investment grows by a factor of 1.06 each year, which corresponds to 6% annual growth. Step 4: Therefore, 2400 represents the initial investment amount, and 1.06 represents the annual growth factor.

8. Sophia is analyzing the growth of an investment in a tech startup. The value of her investment, in thousands of dollars, is modeled by the exponential function V(t) = 9 × (1.07)^t, where t is the number of years since she made the initial investment. Interpret the meaning of the parameters 9 and 1.07 in the context of this investment. Answer: The parameter 9 represents the initial value of the investment in thousands of dollars, so Sophia initially invested $9,000. The parameter 1.07 represents the annual growth factor, meaning the investment increases by 7% each 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 (if b > 1) or decay (if 0 < b < 1) factor per unit time.
   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 (if b > 1) or decay (if 0 < b < 1) factor per unit time. Step 2: In the given function V(t) = 9 * (1.07)^t, compare it to the general form. Here, a = 9 and b = 1.07. Step 3: Interpret a = 9. When t = 0, V(0) = 9 * (1.07)^0 = 9 * 1 = 9. This means the initial value of the investment is 9 thousand dollars, so Sophia invested $9,000. Step 4: Interpret b = 1.07. Since 1.07 > 1, this indicates growth. The factor 1.07 means that each year, the investment is multiplied by 1.07, which is equivalent to a 7% increase (because 1.07 = 1 + 0.07, and 0.07 = 7%). So the investment grows by 7% per year. The answer is: The parameter 9 is the initial investment of $9,000, and the parameter 1.07 is the annual growth factor of 1.07, representing a 7% increase per year.