Question

For the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function: y=12,000+8,000sin(0.628x), where the domain is the years since 1980 and the range is the population of the city.. Over this domain, when does the population reach 18,000? 13,000?

Answer 1

To find when the population reaches 18,000 or 13,000, we can solve the equation y = 12,000 + 8,000sin(0.628x) for x. After solving, we find that the population reaches 18,000 after approximately 2.2229 years and reaches 13,000 after approximately 0.1997 years since 1980.

Step-by-step explanation:

To find when the population reaches 18,000, we need to solve the equation y = 12,000 + 8,000sin(0.628x) for x. We can set y to 18,000 and solve for x, using algebra:

18,000 = 12,000 + 8,000sin(0.628x)

6,000 = 8,000sin(0.628x)

0.75 = sin(0.628x)

Using an inverse trigonometric function, we find that x = arcsin(0.75)/0.628

Using a calculator, we find x ≈ 2.2229

So, the population reaches 18,000 after approximately 2.2229 years since 1980.

To find when the population reaches 13,000, we repeat the same steps and solve the equation y = 12,000 + 8,000sin(0.628x) for x:

13,000 = 12,000 + 8,000sin(0.628x)

1,000 = 8,000sin(0.628x)

0.125 = sin(0.628x)

Using an inverse trigonometric function, we find that x = arcsin(0.125)/0.628

Using a calculator, we find x ≈ 0.1997

So, the population reaches 13,000 after approximately 0.1997 years since 1980.

Answer 2

The answer would be 294762