Really need help with one problem.

The table shows the annual consumption of cheese per person in the United States for selected years in the 19th century. What cubic model best fits this data?

Year Pounds Consumption
1915 4.156
1922 4.857
1946 7.16
1991 29.989

Not sure but do you set it up like this:
ax^3+bx^2+cx^+d=y
a(4.156)^3+b(4.156)^2+c(4.156)+d=1915
a(4.857)^3+b(4.857)^2+c(4.857)+d=1922
a(7.16)^3+b(7.16)^2+c(7.16)+d=1946
a(29.989)^3+b(29.989)^2+c(29.989)+d=1991

Now don't know where to go from here don't have proper calculator or even how to use the online one.

You got it backwards

let y = ax^3 + bx^2 + cx + d
where x is the year, and y is the consumption
you substituted in the wrong variables

let's set 1915 as x=0
1922 -------> x = 7
1946 -------> x = 31
1991 -------> x = 76

so
0 + 0 + 0 + d = 4.156
343a + 49b + 7c + d = 4.857
29791a + 961b + 31c + d = 7.16
438976a + 5776b + 76c + d = 29.989

this would be a horrible system to solve,
I have found a webpage that solves a linear system of equations up to 5by5
http://math.cowpi.com/systemsolver/4x4.html

it gave me
a = 0.00008022
b = -0.00318329
c = 0.1184952015
d = 4.156

so
y = 0.00008022x^3 -0.00318329x^2 +0.1184952015x + 4.156
remember, x is the time since 1915

(using my calculator, I actually tested these values, they work for all 4 equations !! )

To find the cubic model that best fits the data, you need to solve the system of equations you have set up. However, solving this system of equations by hand can be challenging without a proper calculator.

To proceed, you can use an online graphing calculator to solve the system of equations. Here's a step-by-step guide on how to use an online graphing calculator to find the cubic model:

1. Open your web browser and search for "online graphing calculator."

2. Choose one of the available calculators and open it.

3. Enter the equations into the calculator. For example, enter the following equations into the calculator:

- Equation 1: (a * 4.156^3) + (b * 4.156^2) + (c * 4.156) + d = 1915
- Equation 2: (a * 4.857^3) + (b * 4.857^2) + (c * 4.857) + d = 1922
- Equation 3: (a * 7.16^3) + (b * 7.16^2) + (c * 7.16) + d = 1946
- Equation 4: (a * 29.989^3) + (b * 29.989^2) + (c * 29.989) + d = 1991

4. Solve the system of equations using the calculator's solving function. Usually, there is an option to solve systems of equations.

5. After solving, the calculator will provide you with the values of a, b, c, and d, which represent the coefficients of the cubic model that best fits the data.

Alternatively, if you don't have access to an online graphing calculator, you can use mathematical software like MATLAB or Python with libraries such as NumPy and SciPy to solve the system of equations. These software options offer more flexibility and control in solving systems of equations.

Remember to check the documentation or tutorials on how to use the chosen calculator or software if you are unfamiliar with the specific tools.

To find the cubic model that best fits the data, you need to solve the system of equations you set up. We can go through the steps together.

First, let's simplify the equations by raising the values of x to their respective powers:

a(4.156)^3 + b(4.156)^2 + c(4.156) + d = 1915
a(4.857)^3 + b(4.857)^2 + c(4.857) + d = 1922
a(7.16)^3 + b(7.16)^2 + c(7.16) + d = 1946
a(29.989)^3 + b(29.989)^2 + c(29.989) + d = 1991

Now, we have a system of four equations in terms of the unknowns a, b, c, and d. To solve this system, you may need a calculator or access to an online calculator that can handle large calculations. If you don't have access to a calculator, you can try using an online scientific calculator. There are many free options available online.

To solve the system of equations, plug in the values of x and y from each data point into the equations, then use a calculator to solve for a, b, c, and d. Once you have those values, you can write the cubic model equation in the form: y = ax^3 + bx^2 + cx + d.

If you need further assistance, please let me know.