Question

The formula A = 23.1 e 0.01521 models the population of a US state, A, in millions, t years after 2000.
a. What was the population of the state in 2000?
b. When will the population of the state reach 28.3 million?

a. In 2000, the population of the state was__million.
b. The population of the state will reach 28.3 million in ___.

Answer 1

Answer:

- The population in 2001 is 23.1 million

- It will reach 28.3 million in 2013

Step-by-step explanation:

Given

$A =23.1e^{0.0152t}$

Solving (a): Population in 2000.

This implies that

$t = 2000 - 2000$

$t = 0$

So, we have:

$A =23.1e^{0.0152t}$

$A =23.1e^{0.0152*0}$

$A =23.1e^{0}$

Solve the exponent

$A =23.1*1$

$A =23.1$

The population in 2001 is 23.1 million

Solving (b): When will it reach 28.3 million

This implies that:

$A =28.3$

So, we have:

$A =23.1e^{0.0152t}$

$28.3 = 23.1e^{0.0152t}$

Divide both sides by 23.1

$1.225 = e^{0.0152t}$

Take natural logarithm of both sides

$ln\ 1.225 = ln(e^{0.0152t})$

Rewrite as:

$ln\ 1.225 = 0.0152t\ ln(e)$

$ln(e) = 1$

So:

$ln\ 1.225 = 0.0152t*1$

$ln\ 1.225 = 0.0152t$

Make t the subject

$t = \frac{ln\ 1.225}{0.0152}$

$t = \frac{0.2029}{0.0152}$

$t \approx 13$

Add the value of t to 2013 to get the actual year

$Year = 2000 +13$

$Year = 2013$

It will reach 28.3 million in 2013