The population of a town is modelled by P=-0.1x^2+1.2x+4.4 where x is the number of years since the year 2000, and �y is the population in thousands.

a) In the year 2005, what is the population?

I used the equation, put x=5 and I got 7900 as the population.

b) In the year 1999, what is the population?

I used the equation, put x=-1 and I got 3100 as the population.

c) When was the population of the town the greatest? What was the
greatest population?

I used the vertex, (6,8). So in 2006 the greatest population was 8000 people.

d) The town has really become a terrible place to live.Predict when all the residents will leave the town.

How would I do this? Please help

you want P(x) = 0

Just use the quadratic formula to get x = 14.9

so, by the end of 2014 P will have dropped to zero.

To predict when all the residents will leave the town, we need to determine when the population becomes zero.

The population is modelled by the equation P = -0.1x^2 + 1.2x + 4.4, where P represents the population in thousands and x represents the number of years since the year 2000.

To find when the population becomes zero, we can set the equation equal to zero and solve for x:

0 = -0.1x^2 + 1.2x + 4.4

To solve this quadratic equation, we can use the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

In this case, a = -0.1, b = 1.2, and c = 4.4. Substituting these values into the quadratic formula, we have:

x = (-(1.2) ± sqrt((1.2)^2 - 4(-0.1)(4.4))) / (2(-0.1))

Simplifying further, we get:

x = (-1.2 ± sqrt(1.44 + 1.76)) / (-0.2)

x = (-1.2 ± sqrt(3.2)) / (-0.2)

Now we have two possible solutions:

x = (-1.2 + sqrt(3.2)) / (-0.2)

x = (-1.2 - sqrt(3.2)) / (-0.2)

Evaluating these expressions, we find:

x ≈ -0.324 or x ≈ 11.824

Since the given time period is measured in years since the year 2000, a negative value for x does not make sense in this context. Therefore, we discard -0.324 as a valid solution.

The second solution, x ≈ 11.824, corresponds to the year 2012.824. Since it's not realistic to have a fraction of a year, we round up to the nearest whole number to get the predicted year:

The prediction is that all the residents will leave the town around the year 2013.

Please note that this is a mathematical prediction based on the given model, and there may be various factors in reality that can affect the population dynamics of a town.

To answer part (d) of the question, you need to find when the population reaches zero. In the given equation, the population is modeled by the quadratic function P = -0.1x^2 + 1.2x + 4.4.

To find when the population reaches zero, set P = 0 and solve for x:

0 = -0.1x^2 + 1.2x + 4.4

This equation represents a quadratic equation. You can solve it by factoring, completing the square, or by using the quadratic formula. Let's use the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

In this case, a = -0.1, b = 1.2, and c = 4.4. Substituting these values into the quadratic formula:

x = (-1.2 ± sqrt(1.2^2 - 4(-0.1)(4.4))) / 2(-0.1)

Simplifying:

x = (-1.2 ± sqrt(1.44 + 1.76)) / (-0.2)

x = (-1.2 ± sqrt(3.2)) / (-0.2)

To predict when all the residents will leave the town, consider the term within the square root. Since the square root of 3.2 is a positive number, there are real solutions for this equation. However, the negative sign of -1.2 indicates that the population will decline and eventually reach zero.

Therefore, there will be a time in the future when all the residents will leave the town. However, without knowing the context or specific data, it is not possible to determine when that will be. Further information, such as population trends or additional data, would be required for a more accurate prediction.