Find a polynomial function to model the data.

a) f(x) = 4.68x^3 + 8.87x^2 + 4.25x + 6.32

b) f(x) = 4.68x^2 + 8.87x + 6.32

c) f(x) = –4.68x^3 – 8.87x^2 – 4.25x – 6.32

d) f(x) = 0.4468x^3 + 8.87x^2 + 4.25x + 6.32 1

the table shows the number of oak trees planted in tree farms in oregon since 1987. find a cubic function to model the data and use it to estimate the number of oaks planted in 1998.

t(x) = 0.4x^3 + 0.7x^2 + 0.1; 343.5 thousand trees

t(x) = 0.4x^3 + 0.4x^2 + 0.2x + 0.4; 343.5 thousand trees

t(x) = 0.2x^3 + 0.7x^2 + 0.2x + 0.1; 353.2 thousand trees

t(x) = 0.2x^3 + 0.4x^2 + 0.4; 353.2 thousand trees

answer both questions and show or explain answers : )

A) f(x) = 4.68x^3 + 8.87x^2 + 4.25x + 6.32 T(x) = 0.2x^3 + 0.7x^2 + 0.2x + 0.1; 353.2 thousand trees This problem isn't so much as finding or calculating a polynomial function to fit the data as it is in selecting a polynomial function that best fits the data. As it turns out, this is a much easier problem. The usual criteria for a best fit is the least squares method where you attempt to minimize the average of the square of the difference between the predicted points and the actual points. This is easiest done with a spreadsheet, but you can do it manually. Just look at each of the available options and calculate how well it fits the data. I'll go through the entire process for the 1st option, and simply give the results for the 2nd through 4th options. A) f(x) = 4.68x^3 + 8.87x^2 + 4.25x + 6.32 (f(-2.5) - (-24))^2 = (-21.9925 - (-24))^2 = 2.0075^2 = 4.03005625 (f(-2.0) - (-5))^2 = (-4.14 - (-5))^2 = 0.86^2 = 0.7396 (f(-1.5) - 13)^2 = (4.1075 - 13)^2 = -8.8925^2 = 79.07655625 (f(-1.0) - 7)^2 = (6.26 - 7)^2 = -0.74^2 = 0.5476 (f(-0.5) - (-3))^2 = (5.8275 - (-3))^2 = 8.8275^2 = 77.92475625 (f(0.0) - 2)^2 = (6.32 - 2)^2 = 4.32^2 = 18.6624 (f(0.5) - 15)^2 = (11.2475 - 15)^2 = -3.7525^2 = 14.08125625 (f(1.0) - 29)^2 = (24.12 - 29)^2 = -4.88^2 = 23.8144 (f(1.5) - 48)^2 = (48.4475 - 48)^2 = 0.4475^2 = 0.20025625 (f(2.0) - 86)^2 = (87.74 - 86)^2 = 1.74^2 = 3.0276 And the average of 4.03005625, 0.7396, 79.07655625, 0.5476, 77.92475625, 18.6624, 14.08125625, 23.8144, 0.20025625, and 3.0276 is 22.21044813 B) f(x) = 4.68x^2 + 8.87x + 6.32 Average of the squares of the errors is 357.5570325 C) f(x) = –4.68x^3 – 8.87x^2 – 4.25x – 6.32 Average of the squares of the errors is 1138.451948 D) f(x) = 0.4468x^3 + 8.87x^2 + 4.25x + 6.32 Average of the squares of the errors is 794.1515448 And since 22.21044813 is the smallest of the 4 average squared errors calculated, that means that the correct answer is "A". For the second problem, you do exactly the same thing as the first problem. Look at all of the provided functions for the options, determine the error for each function, select the function with the smallest error, and then use f(1998-1987) = f(11) to calculate the estimate. So T(x) = 0.4x^3 + 0.7x^2 + 0.1; Average error = 5170.176 T(x) = 0.4x^3 + 0.4x^2 + 0.2x + 0.4;Average error = 3619.752 T(x) = 0.2x^3 + 0.7x^2 + 0.2x + 0.1;Average error = 0 T(x) = 0.2x^3 + 0.4x^2 + 0.4;Average error = 198.312 And it's pretty obvious that the best fitting function is T(x) = 0.2x^3 + 0.7x^2 + 0.2x + 0.1 with an average error squared of 0. So calculating it with an x of 11, we get 353.2. So the answer is: T(x) = 0.2x^3 + 0.7x^2 + 0.2x + 0.1; 353.2 thousand trees