Suppose that the demand function for a consumer item is given by q = D(p) = 411 - 3p where p is the price in dollars for one item and q is the number of units. If the marginal revenue is 5 dollars per each item that's made and sold, then to the nearest dollar, what is the total revenue that will be generated?
revenue = price * quantity, so
r = pq = (411-q)/3 * q = 1/3 (411q - q^2)
dr/dq = 1/3 (411-2q) = 5
q = (411-15)/2 = 198
so now figure r(198)
To find the total revenue generated, we need to multiply the quantity sold by the price per item.
First, let's find the price per item. The marginal revenue is defined as the additional revenue generated by selling one more item. In this case, the marginal revenue is given as $5.
Since marginal revenue is equal to the slope of the total revenue function, we can say that the slope of the demand function D(p) is also $5. So, we can write:
D'(p) = 5
The demand function is given as q = 411 - 3p. We need to find the value of p that satisfies the above equation. Let's differentiate the demand function with respect to p:
d/dp (411 - 3p) = -3
Setting this equal to 5:
-3 = 5
This equation has no solution, which means there is no value of p that satisfies D'(p) = 5.
Therefore, the given information is inconsistent, and we cannot determine the total revenue generated.