We all need help on this.
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 we do this? Please help
d) the parabola opens downward ... so at some point beyond the vertex , the population (y) will go to zero (one of the "roots")
To determine when all the residents will leave the town, we need to find the x-value when the population (P) becomes zero. This can be done by setting the equation P = -0.1x^2 + 1.2x + 4.4 equal to zero and solving for x.
Let's solve the equation:
-0.1x^2 + 1.2x + 4.4 = 0
Let's use the quadratic formula, where a = -0.1, b = 1.2, and c = 4.4:
x = (-b ± √(b^2 - 4ac)) / (2a)
x = (-1.2 ± √(1.2^2 - 4(-0.1)(4.4))) / (2(-0.1))
Simplifying further:
x = (-1.2 ± √(1.44 + 1.76)) / (-0.2)
x = (-1.2 ± √3.2) / (-0.2)
x = (-1.2 ± √3.2) / (-0.2)
Using the quadratic formula, we get two possible solutions for x:
x ≈ 14.15 or x ≈ -0.95
Since we cannot have a negative number of years, we can ignore the x ≈ -0.95 solution. Thus, the residents will all leave the town after approximately 14.15 years from the year 2000.
To determine the exact year, we can add the approximate value of x to the year 2000:
Year = 2000 + 14.15 ≈ 2014.15
Therefore, it is predicted that all the residents will leave the town around the year 2014.
To predict when all the residents will leave the town, we need to determine when the population reaches zero.
Let's set up the equation:
P = -0.1x^2 + 1.2x + 4.4
Set P = 0 and solve for x:
0 = -0.1x^2 + 1.2x + 4.4
Since this is a quadratic equation, we can solve it by factoring, completing the square, or using the quadratic formula. In this case, the quadratic formula will be the easiest method to use:
x = (-b ± √(b^2 - 4ac)) / (2a)
Using the coefficients from the equation (-0.1x^2 + 1.2x + 4.4), we get:
x = (-(1.2) ± √((1.2)^2 - 4(-0.1)(4.4))) / (2(-0.1))
Simplifying:
x = (-1.2 ± √(1.44 + 1.76)) / (-0.2)
x = (-1.2 ± √3.2) / (-0.2)
Now, we have two values for x: one positive and one negative. The positive value will give us the future date when the population reaches zero since we are measuring time since the year 2000.
Calculating the positive value of x:
x = (-1.2 + √3.2) / (-0.2)
x ≈ 15.65
So, the population is predicted to reach zero in approximately 15.65 years from the year 2000. To determine the year, we add 15.65 years to the year 2000:
Year = 2000 + 15.65 ≈ 2015.65
Therefore, the prediction is that all residents will leave the town around the middle of 2016.