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Pete wants to start an apple selling business. In his business plan, these are the expected expenditures:

Building a cart. $10

Building a big sign. $25

Apples (from farmers). $1 per unit

It is expected that Pete will sell 75% of what he gets from the farmers.

(1) Express the total cost, total revenue, and the profits function in P and Q, where P and Q are the

price Pete charge to his customers and the quantity he sells.

(2) If P= $2, what is O in order to break-even?

Answer :

(1) The profits function can expressed as difference between total revenue and total cost: π(P, Q) = TR(P, Q) - TC(P, Q) and (2) For break-even when selling apples at price of $2, Pete needs to sell a quantity of 35 units (O).

(1) The total cost (TC) function in terms of price (P) and quantity (Q) for Pete's apple selling business can be expressed as:

TC(P, Q) = 10 + 25 + (1 * Q)

The first two terms, $10 for building a cart and $25 for building a big sign, represent the fixed costs, which do not vary with the quantity of apples sold. The third term, (1 * Q), represents the variable cost of acquiring apples from farmers, where the cost per unit is $1 and Q represents the quantity of apples sold.

The total revenue (TR) function is:

TR(P, Q) = P * Q

This function calculates the total revenue generated by selling Q units of apples at price P.

The profits function (Ï€) can be expressed as the difference between total revenue and total cost:

Ï€(P, Q) = TR(P, Q) - TC(P, Q)

(2) If P = $2, in order to break-even (where profits are zero), we need to find the quantity (Q) that satisfies this condition. We set the profits function equal to zero and solve for Q:

0 = TR($2, Q) - TC($2, Q)

To break-even, we need the total revenue to be equal to the total cost. Plugging in the values into the equations, we get:

2Q = 10 + 25 + Q

Q = 35

Therefore, in order to break-even when selling apples at a price of $2, Pete needs to sell a quantity of 35 units (O).

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