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

Question

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

Answer 1

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.

Answer 2

Brilliant, thanks very much for that