Question

The population P of a small town, measured in hundreds, is modeled by the inverse of the function P−1(t) = 10ln(50t − 1000), where t is measured in years. Find the equation to model the population and determine its initial value.
P = 5t +100 where P(0) = 100
P = 1/50 In(0.1t)+20
P = 1/500e^t + 20
P = 1/50e^0.1t + 20

Answer 1

Using the inverse function, it is found that the equation is:

$P(t) = \frac{e^{0.1t}}{50} + 20$

It's initial value is 20.

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We want to find the inverse of:

$y = P^{-1}(t) = 10\ln{50t - 1000}$

To do this, we exchange y and t, and isolate y.

Then:

$10\ln{50y - 1000} = t$

$\ln{50y - 1000} = \frac{t}{10}$

$e^{\ln{50y - 1000}} = e^{0.1t}$

$50y - 1000 = e^{0.1t}$

$50y = e^{0.1t} + 1000$

$y = \frac{e^{0.1t} + 1000}{50}$

$y = \frac{e^{0.1t}}{50} + \frac{1000}{50}$

$P(t) = \frac{e^{0.1t}}{50} + 20$

It's initial value is:

$P(0) = \frac{e^{0.1(0)}}{50} + 20 = 0.02 + 20 = 20.02$

Rounding, 20.

A similar problem is given at brainly.com/question/23950969