Final answer:
The revenue function is obtained by multiplying the price per barrel function by the quantity x, and for 850 thousand barrels, the revenue can be calculated.
Step-by-step explanation
The relationship between the price per barrel of beer (P) and the number of barrels sold annually (x) at the Namibian Breweries is given by the function P = 209.724x⁻¹⁻ˣ, where x is in thousands of barrels. To find the revenue function, we need to multiply the price per barrel by the number of barrels sold (x), which gives us:
R(x) = P × x = (209.724x⁻¹⁻ˣ) × x = 209.724x¹⁻ˣ
When 850,000 barrels of beer are sold, which is 850 in thousands, the revenue R(x) can be computed by plugging x = 850 into the revenue function:
R(850) = 209.724 × 850¹⁻ˣ
To approximate the marginal revenue when 850,000 barrels are sold, we need to take the derivative of the revenue function with respect to x and evaluate it at x = 850.
The change in revenue when production is increased from 850,000 barrels depends on the rate of change of the revenue with respect to the number of barrels sold, which is the marginal revenue.