Answer: a) We know that the Revenue equation is a polynomial of order 1, so we can write it in the form:
R(x) = ax + b
where a and b are constants to be determined. We also know that R(1) = 1 and R(5) = 9, so we have two equations:
a(1) + b = 1
a(5) + b = 9
Solving these equations simultaneously, we get:
a = 2, b = -1
Therefore, the Revenue equation is:
R(x) = 2x - 1
Similarly, the Cost equation is a polynomial of order 2, so we can write it in the form:
C(x) = ax^2 + bx + c
where a, b, and c are constants to be determined. We know that C(1) = R(1) = 1 and C(5) = R(5) = 9, so we have two equations:
a(1)^2 + b(1) + c = 1
a(5)^2 + b(5) + c = 9
We also know that C(2) = 0, so we have a third equation:
a(2)^2 + b(2) + c = 0
Solving these equations simultaneously, we get:
a = 1, b = -4, c = 4
Therefore, the Cost equation is:
C(x) = x^2 - 4x + 4
b) The common solutions to the Revenue and Cost equations represent the production level(s) at which the company makes a profit, since the Revenue equation represents the amount of money the company earns and the Cost equation represents the amount of money the company spends. In other words, the common solutions are the values of x for which R(x) - C(x) > 0. At these production levels, the company's revenue is greater than its cost, resulting in a profit. At production levels where R(x) - C(x) < 0, the company is making a loss. At production levels where R(x) - C(x) = 0, the company is breaking even. Therefore, the common solutions to the two equations signify the production levels at which the company makes a profit or breaks even.
Explanation: