Exponential Parameters

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Answer Key & Explanations

Exponential Parameters · Grade 11 · Worksheet 2

1. Aroha is a conservation officer monitoring the recovery of a rare plant species in a national park. She models the number of plants using the exponential function P(t) = 55(1.07)^t, where t is the number of years since the conservation program began, and P(t) is the number of plants at time t. Interpret the meaning of the parameters 55 and 1.07 in the context of the plant population. Answer: The number 55 represents the initial number of plants at the start of the conservation program (t = 0). The number 1.07 represents the annual growth factor, meaning the plant population increases by 7% each year. Solution: The function is P(t) = 55(1.07)^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, P(0) = 55 * (1.07)^0 = 55 * 1 = 55.
   Full step-by-step solution

   Step 1: The function is P(t) = 55(1.07)^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. Step 2: When t = 0, P(0) = 55 * (1.07)^0 = 55 * 1 = 55. Therefore, the number 55 represents the initial number of plants at the start of the conservation program. Step 3: The base b = 1.07. Since 1.07 is greater than 1, the function represents exponential growth. Each year, the plant population is multiplied by 1.07, meaning it retains 100% of the previous year's population plus an additional 7%. Step 4: The annual percentage increase is 1.07 - 1 = 0.07, which is 7%. So the population increases by 7% each year. Final Answer: The parameter 55 is the initial number of plants at t = 0. The parameter 1.07 is the annual growth factor, meaning the population increases by 7% each year.

2. Mason is monitoring the value of a rare vintage guitar he purchased as an investment. The guitar's value in dollars after t years can be modeled by the exponential function V(t) = 2200(1.07)^t. Interpret the meaning of the parameters 2200 and 1.07 in the context of this investment, and then determine the value of the guitar after exactly 7 years, rounding to the nearest dollar. Answer: 3533 Solution: Interpret the parameters. In V(t) = 2200(1.07)^t, the number 2200 is the initial value a, meaning the guitar was worth $2200 when Mason purchased it (at t = 0 years).
   Full step-by-step solution

   Step 1: Interpret the parameters. In V(t) = 2200(1.07)^t, the number 2200 is the initial value a, meaning the guitar was worth $2200 when Mason purchased it (at t = 0 years). The number 1.07 is the growth factor b, meaning the guitar's value increases by 7% each year (since 1.07 = 1 + 0.07, a 7% annual increase). Step 2: To find the value after 7 years, substitute t = 7 into the function: V(7) = 2200(1.07)^7 Step 3: Calculate (1.07)^7: 1.07^2 = 1.1449 1.07^4 = (1.1449)^2 = 1.31079601 1.07^7 = 1.07^4 * 1.07^2 * 1.07^1 = 1.31079601 * 1.1449 * 1.07 First multiply: 1.31079601 * 1.1449 ≈ 1.5007305 Then multiply: 1.5007305 * 1.07 ≈ 1.6057816 Step 4: Multiply by the initial value: V(7) = 2200 * 1.6057816 ≈ 3532.7195 Step 5: Round to the nearest dollar: 3533 The answer is 3533.

3. 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.

4. The graph of an exponential function of the form f(t) = a * b^t is shown on a coordinate plane. The curve passes through the points (0, 7) and (3, 189). What are the values of a and b, and what do they represent in the context of this function? Answer: a = 7, b = 3 Solution: At t = 0, f(0) = a * b^0 = a * 1 = a. Since the curve passes through (0, 7), we have a = 7. This is the initial value of the function.
   Full step-by-step solution

   Step 1: At t = 0, f(0) = a * b^0 = a * 1 = a. Since the curve passes through (0, 7), we have a = 7. This is the initial value of the function. Step 2: Now substitute the second point (3, 189) into the function with a = 7: f(3) = 7 * b^3 = 189. Step 3: Divide both sides by 7 to isolate b^3: b^3 = 189 / 7 = 27. Step 4: Take the cube root of both sides: b = cube root of 27 = 3. Step 5: Therefore, a = 7 (the initial value when t = 0) and b = 3 (the growth factor, meaning the quantity triples each time unit). The answer is a = 7, b = 3.

5. Sophia is a marine biologist studying the growth of a coral reef. She models the area covered by the coral (in square meters) using the exponential function A(t) = 150(1.08)^t, where t is the number of years since the study began. Interpret the meaning of the parameters 150 and 1.08 in the context of the coral reef's growth. Then, determine the area covered by the coral after 7 years, rounding your answer to the nearest whole square meter. Answer: 257 Solution: Interpret the parameters. The function is A(t) = 150(1.08)^t. The parameter 150 is the initial value a.
   Full step-by-step solution

   Step 1: Interpret the parameters. The function is A(t) = 150(1.08)^t. The parameter 150 is the initial value a. This means at the start of the study (t=0), the coral covered 150 square meters. The parameter 1.08 is the growth factor b. Since 1.08 > 1, it represents a growth rate of 8% per year. Each year, the coral's area is multiplied by 1.08. Step 2: Find the area after 7 years. Substitute t = 7 into the function: A(7) = 150(1.08)^7 Step 3: Calculate (1.08)^7. 1.08^2 = 1.1664 1.08^4 = (1.1664)^2 = 1.36048896 1.08^6 = 1.36048896 * 1.1664 = 1.586874322944 1.08^7 = 1.586874322944 * 1.08 = 1.71382426877952 Step 4: Multiply by 150. A(7) = 150 * 1.71382426877952 = 257.073640316928 Step 5: Round to the nearest whole square meter. 257.0736 rounds to 257. The answer is 257.

6. 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.

7. A radioactive substance decays exponentially. A scientist observes that after 24 hours, only 12.5% of the original sample remains. The decay is modeled by the function N(t) = N₀ * e^(kt), where N(t) is the amount remaining after t hours. Determine the value of the decay constant k. Answer: -0.0866 Solution: N(t) = N₀ * e^(k t) After 24 hours, only 12.5% of the original sample remains.
   Full step-by-step solution

   We are given the exponential decay model: N(t) = N₀ * e^(k t) --- **Step 1: Interpret the problem** After 24 hours, only 12.5% of the original sample remains. That means: N(24) = 0.125 * N₀ --- **Step 2: Substitute into the equation** 0.125 * N₀ = N₀ * e^(k * 24) Divide both sides by N₀: 0.125 = e^(24 k) --- **Step 3: Take natural logarithm of both sides** ln(0.125) = ln(e^(24 k)) ln(0.125) = 24 k --- **Step 4: Evaluate ln(0.125)** 0.125 = 1/8 = 1/(2^3) = 2^(-3) ln(0.125) = ln(2^(-3)) = -3 * ln(2) We know ln(2) ≈ 0.693147 So ln(0.125) ≈ -3 * 0.693147 = -2.079441 --- **Step 5: Solve for k** -2.079441 = 24 k k = -2.079441 / 24 k ≈ -0.086643 --- **Step 6: Round to 4 decimal places** k ≈ -0.0866 --- **Final answer:** k = -0.0866