Suppose that q = D(p) = 800 - 5p is the demand function for a certain consumer item with p as the price in dollars for one unit of this item and q as the number of units. Find the greatest possible revenue by first finding the price p at which the marginal revenue becomes zero and then finding out the revenue that results at this price.
Answer choices:
a) 32000
b)12000
c)54000
d)64000
e)8000
F) none of the above
R = p q = p(800-5p) = 800 p - 5 p^2
dR/dp = 0 when R is max
dR/dp = 800 - 10 p
so max when p = 80
then find q
q = 800 - 5 (80) = 400
so the revenue at max = p q = 80 * 400 = 32,000
To find the price at which the marginal revenue becomes zero, we need to find the derivative of the revenue function.
The marginal revenue (MR) is the derivative of the revenue function (R) with respect to the quantity (q).
Given that q = D(p) = 800 - 5p, we can rewrite it as p = (800 - q)/5.
The revenue function (R) is given by R = pq.
Substituting the expression for p in terms of q into the revenue function, we get:
R = [(800 - q)/5] * q
Now, we can find the derivative of R with respect to q:
dR/dq = (800 - q)/5 - q/5 = (800 - 2q)/5
To find the price at which the marginal revenue becomes zero, we need to solve the equation dR/dq = 0:
(800 - 2q)/5 = 0
800 - 2q = 0
2q = 800
q = 400
Now that we have found q, we can substitute it back into the demand function to find the price p:
p = (800 - q)/5 = (800 - 400)/5 = 400/5 = 80
So, the price p at which the marginal revenue becomes zero is $80.
To find the revenue at this price, we can substitute q = 400 into the revenue function:
R = pq = 80 * 400 = 32000
Therefore, the greatest possible revenue is $32000.
The answer is (a) 32000.
To find the greatest possible revenue, we need to find the price at which the marginal revenue becomes zero.
First, let's determine the revenue function. Revenue is calculated by multiplying the price (p) by the quantity (q):
Revenue (R) = Price (p) * Quantity (q)
Given that the demand function is q = 800 - 5p, we can substitute this expression for q in the revenue equation:
R = p * (800 - 5p)
Next, we need to find the marginal revenue by taking the derivative of the revenue function with respect to price:
Marginal Revenue (MR) = dR/dp
To find the price at which the marginal revenue becomes zero, we set MR = 0 and solve for p:
0 = dR/dp = d(p * (800 - 5p))/dp
To simplify the expression, let's expand and differentiate:
0 = d(800p - 5p^2)/dp
= 800 - 10p
Setting this equal to zero, we have:
800 - 10p = 0
Solving for p:
-10p = -800
p = 80
So, the price at which the marginal revenue becomes zero is p = 80.
Now that we have the price, we can substitute it back into the demand function to find the corresponding quantity:
q = 800 - 5p
= 800 - 5(80)
= 800 - 400
= 400
Therefore, at a price of p = 80, the quantity is q = 400.
Finally, we can calculate the revenue by plugging in the values of p and q into the revenue function:
R = p * q
= 80 * 400
= 32000
Therefore, the greatest possible revenue is 32000.
Hence, the correct answer is a) 32000.