Studies indicate that 𝑡 years from now, the population of a certain country wil be 𝑝 = 0.2𝑡 + 1,500thousand people, and that the gross earnings of the country will be 𝐸 million dollars, where𝐸(𝑡) = √9𝑡2 + 0.5𝑡 + 179a. Express the per capita earning of the country 𝑃 = 𝐸 𝑝⁄ as a function of time 𝑡. (Take care with theunits).b. What happens to the per capita earnings in the long run (𝑎𝑠 𝑡 → ∞).
Question
Studies indicate that 𝑡 years from now, the population of a certain country wil be 𝑝 = 0.2𝑡 + 1,500thousand people, and that the gross earnings of the country will be 𝐸 million dollars, where𝐸(𝑡) = √9𝑡2 + 0.5𝑡 + 179a. Express the per capita earning of the country 𝑃 = 𝐸 𝑝⁄ as a function of time 𝑡. (Take care with theunits).b. What happens to the per capita earnings in the long run (𝑎𝑠 𝑡 → ∞).
Solution
a. To express the per capita earning of the country P = E/p as a function of time t, we first need to express E and p in terms of t.
We know that p = 0.2t + 1500 and E(t) = √(9t^2 + 0.5t + 179).
So, P = E/p = √(9t^2 + 0.5t + 179) / (0.2t + 1500).
This is the per capita earning of the country as a function of time t.
b. To find out what happens to the per capita earnings in the long run (as t → ∞), we need to take the limit of P as t approaches infinity.
As t becomes very large, the term 0.2t in the denominator of P becomes insignificant compared to 1500, and the term 0.5t in the numerator becomes insignificant compared to 9t^2. So, the function P simplifies to √(9t^2) / 1500 = 3t / 1500 = t / 500.
So, as t → ∞, P → ∞. This means that the per capita earnings increase without bound in the long run.
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