Solve the problem.
-The profits (in millions of dollars) of a company can be described by the equation $P(x)=a x^{2}+b x+c$, where $x$ is the number of years since 2000 . Profits were $\$ 7.7$ million in the year 2000, $\$ 9.3$ million in 2004, and $\$ 12.9$ 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.
A)$\$23$ million
B) $\$ 22.1$ million
C) $\$ 23.6$ million
D) $\$ 13.6$ million

Question

Quizplus · Accepted Answer

The correct answer is B) $22.1 million.The given information can be used to set up a system of equations:P(0) = 7.7P(4) = 9.3P(8) = 12.9Substituting the given values into the equation P(x) = ax^2 + bx + c, we get:7.7 = a(0)^2 + b(0) + c9.3 = a(4)^2 + b(4) + c12.9 = a(8)^2 + b(8) + cSimplifying the equations, we get:c = 7.716a + 4b + c = 9.364a + 8b + c = 12.9Substituting c = 7.7 into the second and third equations, we get:16a + 4b = 1.664a + 8b = 5.2Dividing the first equation by 4 and the second equation by 8, we get:4a + b = 0.48a + b = 0.65Subtracting the first equation from the second equation, we get:4a = 0.25a = 0.0625Substituting a = 0.0625 into the first equation, we get:4(0.0625) + b = 0.4b = 0.25Therefore, the equation for the profits is P(x) = 0.0625x^2 + 0.25x + 7.7.To estimate profits in the year 2014, we substitute x = 14 into the equation:P(14) = 0.0625(14)^2 + 0.25(14) + 7.7P(14) = 12.35 + 3.5 + 7.7P(14) = 23.55Rounding to the nearest tenth, the estimated profits in the year 2014 are $23.6 million.

Solve the Problem P(x)=ax2+bx+cP(x)=a x^{2}+b x+cP(x)=ax2+bx+c , Where xxx Is the Number of Years Since 2000

Solve the Problem $P(x)=a x^{2}+b x+c$ , Where $x$ Is the Number of Years Since 2000