The demand function Q and total cost function T(q) of a commodity are given by the equations P=100−4q, and TC=320+4q, where P and q are the price and quantity, respectively. Determine the quantity that must be produced and sold at breaks even.
Question
The demand function Q and total cost function T(q) of a commodity are given by the equations P=100−4q, and TC=320+4q, where P and q are the price and quantity, respectively. Determine the quantity that must be produced and sold at breaks even.
Solution
The break-even point is where total revenue equals total cost.
First, we need to find the total revenue function. The total revenue (TR) is the price (P) times the quantity (q). So, we substitute P from the demand function into the total revenue function:
TR = P*q = (100 - 4q) * q = 100q - 4q^2
Next, we set the total revenue equal to the total cost and solve for q:
100q - 4q^2 = 320 + 4q
Rearrange the equation:
4q^2 - 96q + 320 = 0
Divide the equation by 4 to simplify:
q^2 - 24q + 80 = 0
This is a quadratic equation in the form of ax^2 + bx + c = 0. We can solve for q using the quadratic formula q = [-b ± sqrt(b^2 - 4ac)] / (2a):
q = [24 ± sqrt((24)^2 - 4180)] / (2*1) q = [24 ± sqrt(576 - 320)] / 2 q = [24 ± sqrt(256)] / 2 q = [24 ± 16] / 2 q = 8 or q = 20
So, the quantity that must be produced and sold to break even is either 8 units or 20 units. However, since the demand function is P = 100 - 4q, if q = 20, then P = 0, which is not realistic in a business context. Therefore, the quantity that must be produced and sold to break even is 8 units.
Similar Questions
The demand function Q and cost function C(Q) of a commodity are given by the equations \[Q=20-0.01P,\] and C(Q)=60+6Q, where P and Q are the price and quantity, respectively. The total revenue function (TR) in terms of P is
If the revenue function for a particular commodity is R(p) = −51p2 + 65p, what is the (linear) demand function q(p)?q(p) = Give a reason for your answer.R = p + qR = qp R = q − pR = pqR = pq
ADVANCED ANALYSIS Assume that demand for a commodity is represented by the equation P=18−2Qd.𝑃=18−2𝑄𝑑. Supply is represented by the equation P=−2+2Qs,𝑃=−2+2𝑄𝑠, where Qd and Qs are quantity demanded and quantity supplied, respectively, and P is price.Instructions: Enter your answer for price rounded to 2 decimal places and enter your quantity as a whole number.a. Using the equilibrium condition Qs = Qd, determine equilibrium price.$ .b. Now determine equilibrium quantity. units.
The demand function of a good is given byP = 140 − 35Qwhere P denotes the price and Q the quantity.(a) Write down the revenue R as a function of the quantity Q.(b) Find the quantity Q that maximizes the revenue and find the maximumrevenue
In a market the demand curve is given by P = 100 –2q and supply by P =2q. What are the equilibrium price and quantity traded? Group of answer choicesP* = 25, q* = 25P* = 100, q* = 50P* = 50, q* = 50P* = 50, q* = 25None of the above.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.