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\n(a) what was the population in 1910, according to this model?\n(b) is the graph of this function concave up or concave down? what does this mean?\n(c) use numerical or graphical methods to find when the model estimates the population was 90,560,000\n(a) according to the model, the population in 1910 was 92,201 thousands\n(round to the nearest integer as needed)\n(b) is the graph of this function concave up or concave down?\nconcave up\nconcave down\nwhat does this mean?\nit means that the function is at rate.

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\n(a) what was the population in 1910, according to this model?\n(b) is the graph of this function concave up or concave down? what does this mean?\n(c) use numerical or graphical methods to find when the model estimates the population was 90,560,000\n(a) according to the model, the population in 1910 was 92,201 thousands\n(round to the nearest integer as needed)\n(b) is the graph of this function concave up or concave down?\nconcave up\nconcave down\nwhat does this mean?\nit means that the function is at rate.

Answer

Explanation:

Step1: Analyze the function (y = 165.6x^{1.345})

For a function (y=ax^{n}), the second - derivative (y'') determines concavity. First, find the first derivative using the power rule (y'=nax^{n - 1}). Here, (y'=1.345\times165.6x^{1.345-1}=222.744x^{0.345}).

Step2: Find the second derivative

Using the power rule again, (y''=(n - 1)nax^{n - 2}). So, (y''=0.345\times222.744x^{0.345 - 1}=76.84668x^{-0.655}=\frac{76.84668}{x^{0.655}}). Since (x>0) (as (x) represents the number of years after 1800) and (y''>0) for (x > 0), the function is concave up.

Step3: Interpret concavity

A concave - up function means that the rate of change of the function (the first derivative (y')) is increasing.

Answer:

It means that the function is increasing at an increasing rate.