A low-projection scenario for the population of Smalltown, USA for 1995 and 2150

Question

A low-projection scenario for the population of Smalltown, USA for 1995 and 2150 is given by the function y = -0.35x^2 + 39.15x + 5811.08, where x is the number of years after 1990 and the population is measured in thousands of people. a. Find the input and output at the vertex of the graph of this model. b. Interpret the values from part a. c. For what years after 1995 does this model predict that the population will increase? The at the vertex is x almostequalto and at the vertex is y almostequalto (Round to the nearest hundredth as needed.)

Explanation / Answer

The function projecting the populationof Small town,USA is y = -0.35x2+39.15x +5811.08 = -0.35( x2 – 7.83x/0.07) + 5811.08 = -0.35[ x2 – 2* 7.83/0.14 + (7.83/0.14)2] + 5811.08 +0.35*(7.83/0.14)2 = -0.35( x- 7.83/0.14)2 + 5811.08 + 1094.80 = -0.35( x- 55.93)2 + 6905.88.

This is the equation of a parabola opening downwards. The vertex of the parabola is ( 55.93, 6905.88).

(a). The input at the vertex of the given model is 55.93, say 56 ( on rounding off to the nearest whole number). The output is 6905.88.

(b) As per the graph of the model, the populationof Small town,USA will be maximum in the year 56 years after 1990, i.e. 51 years after 1995 i.e. in the year 2046. The maximum population of this town will mbe 6905.88 thousand.

( c) Since the vertex is the highest point of a parabola opening downwards, the population of Small town,USA will increase for 56 years after 1990, i.e. from 1995 to 2046.

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