Question

-1. The table shows the annual consumption of cheese per person in the United States for selected years in the 20th century. Let x = number of years since 1900, and y = pounds per person.

Year Pounds Consumed
1912 4.976
1932 8.717
1969 16.997
1979 21.705

What cubic model best fits this data?

-2. The table shows the annual consumption of cheese per person in the U.S. for selected years in the 20th century.

Year Pounds Consumed
1913 4.468
1926 5.56
1955 4.778
1990 10.905

Use a cubic model to estimate milk production in 1989.
a. 20.9
b. 10.5
c. 14.5
d. 6.5

-3. Use synthetic division to find P(-1) for P(x)=x^4+x^3+4x^2-10x+4
a. -1
b. -16
c. 18
d. 8

Answer 1

To determine the best cubic model for cheese consumption data, regression analysis with statistical software is needed which we cannot perform here. For the estimation of milk production in 1989, without the specific model, we cannot precisely predict, but likely choices are options (b) 10.5 or (c) 14.5. P(-1) for the polynomial is found to be 18 using synthetic division.

Step-by-step explanation:

To find the cubic model that best fits the data for the annual consumption of cheese per person in the United States, we would generally use a statistical software or graphing calculator to perform a regression analysis. Since we do not have such tools here, we cannot provide the exact cubic model. However, the process involves plotting the given data points, (x, y), where x is the number of years since 1900 and y is the pounds of cheese consumed, and then using the cubic regression feature to generate the model.

For the second question, to estimate milk production in 1989 using a cubic model, we would first determine the cubic equation similarly. Since we are asked to estimate for 1989, which corresponds to x=89, we would substitute this into our cubic equation to find the value of y. Without the specific cubic model, we cannot accurately predict the milk production. However, option (d) 6.5 is clearly too low considering the trend of increasing cheese consumption, while (a) 20.9 might be too high considering the actual value for 1990 was 10.905. Therefore, choices (b) 10.5 or (c) 14.5 seem more plausible.

To find P(-1) for the polynomial P(x) = x^4 + x^3 + 4x^2 - 10x + 4 using synthetic division, we set up the division and find the value:

• -1 | 1 1 4 -10 4
• ----- |------------------
• ------| -1 0 -4 14
• ------| 1 0 4 -14 18

The remainder is 18, therefore P(-1) = 18, which corresponds to option (c) 18.