the u.s. population can be modeled by the function ( y = 165.6x^{1.345} ), where ( y ) is in thousands and (…

the u.s. population can be modeled by the function ( y = 165.6x^{1.345} ), where ( y ) is in thousands and ( x ) is the number of years after 1800. (a) what was the population in 1910, according to this model? (b) is the graph of this function concave up or concave down? what does this mean? (c) use numerical or graphical methods to find when the model estimates the population was 90,560,000. (a) according to the model, the population in 1910 was 92.201 thousands (round to the nearest integer as needed.) (b) is the graph of this function concave up or concave down? concave up concave down

the u.s. population can be modeled by the function ( y = 165.6x^{1.345} ), where ( y ) is in thousands and ( x ) is the number of years after 1800. (a) what was the population in 1910, according to this model? (b) is the graph of this function concave up or concave down? what does this mean? (c) use numerical or graphical methods to find when the model estimates the population was 90,560,000. (a) according to the model, the population in 1910 was 92.201 thousands (round to the nearest integer as needed.) (b) is the graph of this function concave up or concave down? concave up concave down

Answer

Explanation:

Step1: Find the second - derivative

Given (y = 165.6x^{1.345}). First, find the first - derivative using the power rule (y^\prime=\frac{d}{dx}(ax^n)=anx^{n - 1}). (y^\prime=165.6\times1.345x^{1.345-1}=222.732x^{0.345}). Then, find the second - derivative (y^{\prime\prime}=\frac{d}{dx}(222.732x^{0.345})). Using the power rule again, (y^{\prime\prime}=222.732\times0.345x^{0.345 - 1}=76.84254x^{-0.655}=\frac{76.84254}{x^{0.655}}). Since (x>0) (because (x) represents the number of years after 1800) and (y^{\prime\prime}>0) for (x > 0).

Step2: Interpret the concavity

If (y^{\prime\prime}>0) on an interval, the function (y = f(x)) is concave up on that interval. A concave - up graph of the population function (y = 165.6x^{1.345}) means that the rate of growth of the population is increasing over time.

Answer:

Concave up. A concave - up graph of the population function means that the rate of growth of the population is increasing over time.