The management of Lorimar Watch Company has determined that the daily marginal revenue function associated with producing and selling their travel clocks is given by
R '(x) = -0.003x + 12
where x denotes the number of units produced and sold and R '(x) is measured in dollars/unit.
(a) Determine the revenue function R(x) associated with producing and selling these clocks.
R(x) =
(b) What is the demand equation that relates the wholesale unit price p with the quantity of travel clocks demanded?
p =
To determine the revenue function R(x), we need to integrate the given marginal revenue function R'(x) with respect to x.
(a) To integrate R'(x), we treat -0.003x + 12 as a polynomial function and apply the power rule of integration:
∫(-0.003x + 12) dx = -0.003∫xdx + 12∫dx
Using the power rule, we get:
-0.003 * (1/2)x^2 + 12 * x + C
Simplifying, we have:
-0.0015x^2 + 12x + C
Since we are looking for the revenue function R(x), we can ignore the constant of integration C as it doesn't affect the revenue expression. Therefore, the revenue function R(x) associated with producing and selling these clocks is:
R(x) = -0.0015x^2 + 12x
(b) To find the demand equation that relates the wholesale unit price p with the quantity of travel clocks demanded, we need to recall the relationship between marginal revenue and marginal cost.
In economics, the marginal revenue (MR) is equal to the marginal cost (MC) at the point of maximum total revenue. This is because at the point where MR and MC intersect, the additional revenue gained from selling an additional unit equals the additional cost incurred in producing that unit.
In this case, we can assume that the marginal cost is constant and equal to the wholesale unit price p. Therefore, we can equate the given marginal revenue R'(x) to the wholesale unit price p:
R'(x) = p
Substituting the given marginal revenue function:
-0.003x + 12 = p
Rearranging the equation to solve for p:
p = -0.003x + 12
So, the demand equation that relates the wholesale unit price p with the quantity of travel clocks demanded is:
p = -0.003x + 12