The problem provides the total cost function $C(x) = 35x + 1925$ and the total revenue function $R(x) = 70x$ for a ceiling fan manufacturer. We are asked to find: (a) The equation of the profit function $P(x)$. (b) The profit when 35 units are sold, $P(35)$. (c) The number of fans that must be sold to avoid losing money (i.e., to break even or make a profit).

1. Problem Description

The problem provides the total cost function $C(x) = 35x + 1925$ and the total revenue function $R(x) = 70x$ for a ceiling fan manufacturer. We are asked to find:

(a) The equation of the profit function $P(x)$.

(b) The profit when 35 units are sold, $P(35)$.

(c) The number of fans that must be sold to avoid losing money (i.e., to break even or make a profit).

2. Solution Steps

(a) The profit function $P(x)$ is the difference between the revenue function $R(x)$ and the cost function $C(x)$:

$P(x) = R(x) - C(x)$

$P(x) = (70x) - (35x + 1925)$

$P(x) = 70x - 35x - 1925$

$P(x) = 35x - 1925$

(b) To find the profit when 35 units are sold, we substitute $x = 35$ into the profit function $P(x)$:

$P(35) = 35(35) - 1925$

$P(35) = 1225 - 1925$

$P(35) = -700$

(c) To avoid losing money, the profit must be greater than or equal to zero, $P(x) \ge 0$. We need to find the value of $x$ that satisfies this condition:

$35x - 1925 \ge 0$

$35x \ge 1925$

$x \ge \frac{1925}{35}$

$x \ge 55$

Therefore, at least 55 fans must be sold to avoid losing money.

3. Final Answer

(a) $P(x) = 35x - 1925$

(b) $P(35) = -700$

(c) 55 fans