A fish population is modeled by P(t) = 1200 × (1.15)^t, where t is the number of years since 2020. a) Predict…

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A fish population is modeled by P(t) = 1200 × (1.15)^t, where t is the number of years since 2020. a) Predict the population in 2025. b) Show this is equivalent to the log model log P = log 1200 + t l

ScanSolve AI Tutor · Accepted Answer

a) To calculate the predicted fish population in the year 2025 using the index model, we first determine the number of years t from the initial year 2020. Step 1: Calculate the number of years t. The initial year is 2020, and the target year is 2025. t = 2025 − 2020 = 5 years; Step 2: Substitute t=5 into the index model P(t) = 1200 × (1.15)^t. P(5) = 1200 × (1.15)⁵; P(5) = 1200 × 2.0113571875; P(5) = 2413.628625 Rounding to the nearest whole fish, the predicted population is 2414 fish. 2414 fish b) To show that the logarithmic model is equivalent to the index model, we start with the index model and apply the logarithm. Step 1: Start with the index model. P = 1200 × (1.15)^t Step 2: Apply log₁₀ to both sides of the equation. log₁₀ P = log₁₀ (1200 × (1.15)^t); Step 3: Use the logarithm properties log(AB) = log A + log B and log(A^B) = B log A. log₁₀ P = log₁₀ 1200 + log₁₀ (1.15)^t; log₁₀ P = log₁₀ 1200 + t log₁₀ 1.15 This matches the given logarithmic model, thus showing their equivalence. c) To determine the approximate number of years (t) it will take for the fish population to reach 5,000, we use the index model and solve for t using logarithms. Step 1: Set P(t) = 5000 in the index model. 5000 = 1200 × (1.15)^t Step 2: Isolate the exponential term. 5000/1200 = (1.15)^t; 50/12 = (1.15)^t; 25/6 = (1.15)^t Step 3: Take the logarithm (base 10 or natural logarithm) of both sides. log₁₀(25/6) = log₁₀((1.15)^t); Step 4: Use the logarithm property log(A^B) = B log A to solve for t. log₁₀(25/6) = t log₁₀(1.15); t = (log₁₀(25/6))/(log₁₀(1.15)) t ≈ 0.619789/0.060698 t ≈ 10.21 The approximate number of years is 10.21 years. 10.21 years

A fish population is modeled by P(t) = 1200 * (1.15)^t, where t is the number of years since 2020. a) Predict the population in 2025. b) Show this is equivalent to the log model log P = log 1200 + t l

Answers

Step-by-step solution

Part a· 2 steps

Calculate the number of years $t$ .

Substitute $t=5$ into the index model $P(t) = 1200 \times (1.15)^t$ .

Part b· 3 steps

Start with the index model.

Apply $\log_{10}$ to both sides of the equation.

Use the logarithm properties $\log(AB) = \log A + \log B$ and $\log(A^B) = B \log A$ .

Part c· 4 steps
$\text{10.21 years}$

Set $P(t) = 5000$ in the index model.

Isolate the exponential term.

Take the logarithm (base 10 or natural logarithm) of both sides.

Use the logarithm property $\log(A^B) = B \log A$ to solve for $t$ .

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