At the beginning of 2010, City A had a population of 2,425 and was growing by 27 residents per year. City B had a population of 1,813 and was growing by 63 residents per year. If this trend continues in what year will the populations of the two cities be the same?

17 years

Step-by-step explanation:

City A is currently 2,425 people in number and growing at 27 per year.

An expression to find the population in a given year assuming the year is x would be:

= 2,425 + 27 * x

= 2,425 + 27x

Applying the same logic to City B would give an expression of:

= 1,813 + 63x

Equating these together will give the year they will be the same:

2,425 + 27x = 1,813 + 63x

2,425 - 1,813 = 63x - 27x

612 = 36x

x = 612/36

x = 17 years

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.