Question

Hello I need help pleasePart b: According to the model, in what year an amount of 2855 thousand students will be enrolled in schools?

Answer 1

To answer this question, we need to compare what we have from the graph compared with the model given by the function:

`f(x)=15x^2+320`

We know that the function, f(x), represents "the number of students enrolled in schools, in thousands, x years after 2000."

We can also see from the graph that the number of students enrolled is also in thousands.

Then, we have:

1. The number of years after 2000 for 2006 is (2006 - 2000) = 6. This is the number we are going to plug into the equation to have the estimate of the model. Then, we have:

`f(x)=15x^2+320\Rightarrow f(6)=15(6)^2+320\Rightarrow f(6)=15\cdot36+320`
`f(6)=540+320\Rightarrow f(6)=860`

That is, the function of the model, the number of students, in thousands, that were enrolled in 2006 is 860 thousand of students.

According to the graph, we have 835, then the function of the model overestimates the number of students in (860 - 835 = 25), that is, 25 thousand students.

To answer the question: According to the model, in what year an amount of 2855 thousand students will be enrolled in schools? We can proceed as follows:

1. We can use the function of the model, and then equate it to 2855:

`2855=15x^2+320`

2. Subtract 320 to both sides of the equation:

`2855-320=15x^2+320-320\Rightarrow2535=15x^2`

3. Divide both sides of the equation by 15:

`(2535)/(15)=(15)/(15)x^2\Rightarrow x^2=169`

4. Extract the square root to both sides of the equation:

`\sqrt[]{x^2}=\pm\sqrt[]{169}\Rightarrow x=\pm13`

The negative value of x = -13 is meaningless here since we are finding the number of years after 2000, then x = 13.

Therefore, since the model is for the number of students enrolled in schools, in thousands, x years after 2000, then 13 years after 2000 is (2000 + 13 = 2013).

Therefore, in the year 2013, there will be 2855 thousand students enrolled in schools.

In summary, we have that, according to the model, there were 860 thousand students enrolled in schools in 2006, and also in the year 2013, there will be 2855 thousand students enrolled in schools.