Question

The formula A = 23.1 e 0.01521 models the population of a US state, A, in millions, t years after 2000.

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