It is estimated that an annual advertising revenue received by a certain newspaper will be R(x)=0.5x^2+3x+160 thousand dollars when its circulation is x thousand. The circulation of the paper is currently 10,000 and is increasing at a rate of 2,000 per year. At what rate will the annual advertising revenue be increasing with respect to time 2 years from now?

To find the rate at which the annual advertising revenue will increase with respect to time 2 years from now, we need to take the derivative of the revenue function and evaluate it when the circulation is 12,000 (10,000 + 2,000 * 2).

Here's how to do it step by step:

1. Start with the revenue function: R(x) = 0.5x^2 + 3x + 160 (in thousands of dollars).

2. Take the derivative of the revenue function R(x) with respect to x to find the rate at which the revenue is changing with respect to circulation:
R'(x) = dR/dx = d(0.5x^2 + 3x + 160)/dx
= 0.5*2x + 3
= x + 3

3. Evaluate the derivative R'(x) at x = 12,000 (two years from now):
R'(12) = 12,000 + 3
= 12,003

Therefore, the annual advertising revenue will be increasing at a rate of 12,003 thousand dollars per year (or $12.003 million) two years from now.