TC=3q^2-10Q+50
Calculate profit maximization in the short run
To calculate profit maximization in the short run, we need to find the quantity that maximizes total revenue and deduct the total cost associated with that quantity.
Total revenue (TR) can be calculated by multiplying the quantity (q) by the price (p): TR = p * q.
Total cost (TC) is given by the equation TC = 3q^2 - 10q + 50.
Therefore, in order to find the profit-maximizing quantity, we need to determine the quantity (q) that maximizes TR - TC.
Profit (π) is defined as TR - TC: π = TR - TC.
To find the profit-maximizing quantity, we'll differentiate the profit equation with respect to q, set it equal to zero, and solve for q.
π = TR - TC
π = p * q - (3q^2 - 10q + 50)
π = p * q - 3q^2 + 10q - 50
Now, let's differentiate π with respect to q and set it equal to zero:
dπ/dq = p - 6q + 10 = 0
Solving for q:
6q = p + 10
q = (p + 10)/6
Therefore, the profit-maximizing quantity in the short run is (p + 10)/6.