the total revenue function for a product is given by ( r = 520x ) dollars, and the total cost function for…

the total revenue function for a product is given by ( r = 520x ) dollars, and the total cost function for this same product is given by ( c = 16,000 + 80x + x^{2} ), where ( c ) is measured in dollars. for both functions, the input ( x ) is the number of units produced and sold. a. form the profit function for this product from the two given functions. b. what is the profit when 34 units are produced and sold? c. what is the profit when 48 units are produced and sold? d. how many units must be sold to break even on this product?

the total revenue function for a product is given by ( r = 520x ) dollars, and the total cost function for this same product is given by ( c = 16,000 + 80x + x^{2} ), where ( c ) is measured in dollars. for both functions, the input ( x ) is the number of units produced and sold. a. form the profit function for this product from the two given functions. b. what is the profit when 34 units are produced and sold? c. what is the profit when 48 units are produced and sold? d. how many units must be sold to break even on this product?

Answer

Explanation:

Step1: Recall the profit formula

Profit ( P(x)=R(x)-C(x) ). Given ( R(x) = 520x ) and ( C(x)=16000 + 80x+x^{2} ).

Step2: Substitute ( R(x) ) and ( C(x) ) into the profit formula

[ \begin{align*} P(x)&=520x-(16000 + 80x+x^{2})\ &=520x-16000 - 80x-x^{2}\ &=-x^{2}+(520x - 80x)-16000\ &=-x^{2}+440x-16000 \end{align*} ]

Answer:

( P(x)=-x^{2}+440x - 16000 )