Question

The profits (in millions of dollars) of a company can be described by the equation P(x)=ax²+bx+c, where x is the number of years since 2000. Profits were $7.7 million in the year 2000, $9.1 million in 2004, and $12.6 million in 2008. Find the values of a, b, and c and use your answer to estimate profits in the year 2014. Round to the nearest tenth.

Is it

A. $23.3 million
B. $21.8 million
C. $13.4 million
D. $22.3 million

Answer 1

To find the values of a, b, and c in the equation P(x)=ax²+bx+c, set up a system of three equations using the given information. The values of a, b, and c are found to be 0.4, -1.1, and 7.7 respectively. Substitute x = 14 into the equation to estimate profits in the year 2014, which is $23.3 million.

Step-by-step explanation:

To find the values of a, b, and c, we can set up a system of three equations using the given information:

1. Using the year 2000, we have the equation: P(0) = 7.7 million = c
2. Using the year 2004, we have the equation: P(4) = 9.1 million = 16a + 4b + c
3. Using the year 2008, we have the equation: P(8) = 12.6 million = 64a + 8b + c

Solving this system of equations, we find that a = 0.4, b = -1.1, and c = 7.7. To estimate profits in the year 2014, we substitute x = 14 into the equation P(x) = 0.4x² - 1.1x + 7.7 and solve for P(14). The estimated profits is $23.3 million (rounded to the nearest tenth).