According to projections by a organization of countries, the population of a certain country (in billions) can be approximated by the following function where x=0 corresponds to the year 1990. In what year will the country reach its maximum population, and what will that population be?
-0.000177x+0.014595x+1.1386
Check your function. as a linear function, it forever gets bigger.
perhaps you meant:
f(x) = -0.000177x^2 + 0.014595x + 1.1386
in that case find the vertex of this quadratic using the method you learned.
I trust that by now you have
(a) found the vertex at (41.229,1.4395)
(b) know how to use that to answer the question
To find the year when the country will reach its maximum population and the corresponding population, we need to understand that the given function describes the population growth over time.
The formula for the population of the country (in billions) at a given year (x) is:
P(x) = -0.000177x^2 + 0.014595x + 1.1386
To determine when the country will reach its maximum population, we need to find the x-value that corresponds to the vertex of the quadratic function. The vertex of a quadratic function can be found using the formula:
x = -b / (2a)
In this case, a = -0.000177 and b = 0.014595.
Plugging these values into the formula, we get:
x = -0.014595 / (2*(-0.000177))
Simplifying,
x = -0.014595 / (-0.000354)
x ≈ 41.19
This means that the country will reach its maximum population approximately 41.19 years after 1990. To determine the year, we add this value to 1990:
Year = 1990 + 41.19
Year ≈ 2031.19
So, the country is projected to reach its maximum population around the year 2031.19.
To calculate the maximum population itself, we substitute this x-value back into the original equation:
P(x) = -0.000177x^2 + 0.014595x + 1.1386
P(41.19) ≈ -0.000177(41.19)^2 + 0.014595(41.19) + 1.1386
P(41.19) ≈ 1.435
Therefore, the maximum projected population of the country would be approximately 1.435 billion.