Final answer:
The revenue function, based on the given demand equation, is R(q) = 29q - 0.01q^2. The quantity that maximizes revenue is 1450 units. At this production level, the price per item is $14.50, and the total revenue at this price is $42,025.
Step-by-step explanation:
The demand equation for a product is given by p = 29 - 0.01q, where p is the price in dollars and q is the quantity of products sold. To write the revenue function R(q), we multiply the price p by the quantity q, hence:
R(q) = p × q = (29 - 0.01q) × q = 29q - 0.01q2.
To find the quantity q that maximizes revenue, we need to find the vertex of the parabola, since the coefficient of the q2 term is negative. The vertex formula for a parabola y = ax2 + bx + c is x = -b/(2a), and in this case:
q = -(-29)/(2 × (-0.01)) = 1450.
Now we find the price at this quantity level:
p = 29 - 0.01(1450) = 29 - 14.5 = $14.50.
The total revenue at this price is:
R(1450) = 29(1450) - 0.01(14502) = $42,025.