Jimmy decides to mow lawns to earn money. The initial cost of his electric lawnmower is \( \$ 350 \). Electricity and maintenance costs are \( \$ 6 \) per lawn. Complete parts (a) through (c). a) Formulate a function \( C(x) \) for the total cost of mowing \( x \) lawns. b) Jimmy determines that the total-profit function for the lawn mowing business is given by \( P(x)=9 x-350 \). Find a function for the total revenue from mowing \( x \) lawns. \( R(x)=15 x \) How much does Jimmy charge per lawn? \( \$ 15 \) c) How many lawns must Jimmy mow before he begins making a profit? (Round to the nearest integer as needed.)

To solve the problem, let's go through each part step by step. ### Part (a) We need to formulate a function \( C(x) \) for the total cost of mowing \( x \) lawns. The total cost consists of the initial cost of the lawnmower and the variable costs associated with mowing each lawn. 1. **Initial cost of the lawnmower**: \( \$350 \) 2. **Variable cost per lawn**: \( \$6 \) The total cost function \( C(x) \) can be expressed as: \[ C(x) = 350 + 6x \] ### Part (b) We are given the total profit function \( P(x) = 9x - 350 \). Profit is defined as total revenue minus total cost, so we can express this relationship as: \[ P(x) = R(x) - C(x) \] We know that: - \( C(x) = 350 + 6x \) - Rearranging the profit equation gives us: \[ R(x) = P(x) + C(x) \] Substituting the expressions for \( P(x) \) and \( C(x) \): \[ R(x) = (9x - 350) + (350 + 6x) \] \[ R(x) = 9x - 350 + 350 + 6x \] \[ R(x) = 15x \] Thus, the total revenue function is: \[ R(x) = 15x \] From the information given, we can also conclude that Jimmy charges \( \$15 \) per lawn. ### Part (c) To find out how many lawns Jimmy must mow before he begins making a profit, we need to determine when the profit function \( P(x) \) is greater than zero: \[ P(x) = 9x - 350 > 0 \] Solving for \( x \): \[ 9x > 350 \] \[ x > \frac{350}{9} \] \[ x > 38.888\ldots \] Rounding to the nearest integer, Jimmy must mow at least: \[ x = 39 \] Thus, Jimmy must mow **39 lawns** before he begins making a profit.