Question

PLEASE HELP!!! The population P of a small town, measured in hundreds, is modeled by the inverse of the function P−1(t) = 20ln(40t − 1200), where t is measured in years. Find the equation to model the population and determine its initial value.

A. P=2t+60 where P(0)=60
B. P=$\frac{1}{40}e^{0.05t}+30$
C. $P=\frac{1}{40}In(0.05t)+30$
D. $P=\frac{1}{800}e^{t} +30$

(no links pls)

Answer 1

Answer:

B

Step-by-step explanation:

I got it correct on the test.

Answer 2

The equation to model the population is $P(t)=\frac{1}{40}e^{0.05t}+30$ and its initial value is P(0) = 30

The correct answer is option (B)

What is equation?

"It is a mathematical statement which consists of equal symbol between two algebraic expressions."

What is inverse function?

"If f(x) produces y, then putting y into the inverse of f produces the output x."

For given question,

The population P of a small town, measured in hundreds, is modeled by the inverse of the function $P^{-1}(t)=20ln(40t-1200)$,

where t is measured in years.

We need to find an equation to model the population.

Let, $y=P^{-1}(t)$

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

So, we get an equation,

$\Rightarrow 20ln(40y-1200)=t\\\\\Rightarrow ln(40y-1200)=\frac{t}{20}\\\\ \Rightarrow e^{ln(40y-1200)}=e^{0.05t}\\\\\Rightarrow 40y-1200=e^{0..05t}\\\\\Rightarrow 40y=e^{0.05t}+1200\\\\\Rightarrow y=\frac{1}{40}e^{0.05t}+\frac{1}{40}1200\\\\ \Rightarrow y=\frac{1}{40}e^{0.05t}+30\\\\ \Rightarrow P(t)=\frac{1}{40}e^{0.05t}+30$

Now, we determine the initial value.

$\Rightarrow P(0)=\frac{1}{40}e^{0.05\times 0}+30\\\\\Rightarrow P(0)=\frac{1}{40}(1)+30\\\\\Rightarrow P(0)=30.025\\\\\Rightarrow P(0)\approx 30$

Therefore, the equation to model the population is $P(t)=\frac{1}{40}e^{0.05t}+30$ and its initial value is P(0) = 30

The correct answer is option (B)

Learn more about the inverse of the function here:

brainly.com/question/23950969

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