Hi, I have a formula for modelling the variation in population density in terms of distance from a town centre;

y= -3x^2 +6x+9

where y represents the population density (in thousands per km^2) and x represents the distance (in km) from the town centre

I have already constructed a parabola as part of the question its just the next bit I need some help on if possible?

When x = 2 and y =9 what does this mean in th context of the model?

How would i calculate the population density at the town centre?

What is the max population density predicted by the model and at what distance from the town centre would it occur?

For what range of distances fro the town centre might this model be useful?

Any help or guidance would be very much appreciated

when x=2 and y=9

would mean that at a distance of 2 km form the town centre the population density is 9 thousand per km^2

at the town centre, x = 0, so
y = 0+0+9 or 9000 per km^2

The max density would be at the vertex of your parabola.
A quick way to find the vertex:
for y = ax^2 + bx + c, the x value of the vertex is -b/(2a)
so for yours, x = -6/-6 = 1
when x=1, y = -3+6+9 = 12

so 1 km from the centre, the density is 12000/km^2

Since y is your density, and density is obviously a positive number, we would use your parabola only when it is above the x-axis

so let's find the x-intercepts

-3x^2 + 6x + 9 = 0
x^2 - 2x - 3 = 0
(x-3)(x+1) = 0
x = 3 or x = -1

but x = -1 makes little sense here, since clearly x > 0

so your model would only be useful for
x between 0 and 3, and the range of density would be from 0 to 12.

Brilliant, thanks very much for that

Sure, I'd be happy to help you with your questions!

First, let's address the meaning of the values x = 2 and y = 9 in the context of the model. In the given formula y = -3x^2 + 6x + 9, y represents the population density in thousands per km^2, and x represents the distance from the town centre in kilometers. So when x = 2 and y = 9, it means that at a distance of 2 kilometers from the town centre, the population density is 9 thousand people per km^2.

To calculate the population density at the town centre, you need to determine the value of y when x = 0. So, plug in x = 0 into the formula: y = -3(0)^2 + 6(0) + 9. This simplifies to y = 9. Therefore, according to the model, the population density at the town centre is 9 thousand people per km^2.

To find the maximum population density predicted by the model and the distance from the town centre where it occurs, you can use calculus. Start by taking the derivative of the formula with respect to x: y' = -6x + 6. Set the derivative equal to 0 to find the critical points: -6x + 6 = 0. Solve for x: x = 1. Therefore, the maximum population density occurs when x = 1 kilometer from the town centre.

To find the maximum population density itself, substitute x = 1 back into the formula: y = -3(1)^2 + 6(1) + 9. This simplifies to y = 12. So, the model predicts the maximum population density to be 12 thousand people per km^2, and it occurs at a distance of 1 kilometer from the town centre.

As for the range of distances for which this model might be useful, it depends on the context and the specific characteristics of the town. Generally, this model assumes a parabolic shape for the population density variation and may be most accurate within a certain proximity to the town centre. However, beyond a certain distance, other factors may come into play that could affect population density differently. Therefore, it's important to consider the specific characteristics of the town and the reliability of the data and assumptions used to construct the model.

I hope this helps! Let me know if you have any further questions or if there's anything else I can assist you with.