A food truck gyro vendor has weekly fixed costs of $\$ 560$, and variable costs of $\$ 5.30$ for each gyro combo prepared. Complete parts a through c below. a. Let $x$ represent the number of gyro combos prepared and sold each week. Write the weekly cost function, C, for the food truck vendor. (Hint: The cost function is the sum of fixed and variable costs.) \[ C(x)=560+5.30 x \] (Use integers or decimals for any numbers in the expression.) b. The function $R(x)=-0.001 x^{2}+8.54 x$ describes the money, in dollars, that the food truck vendor takes in each week from the sale of $x$ gyro combos. Use this revenue function and the cost function from part (a) to write the vendor's weekly profit function, P. (Hint: The profit is the difference between the revenue and the cost functions.) \[ P(x)=-0.001 x^{2}+3.24 x-560 \] (Simplify your answer. Use integers or decimals for any numbers in the expression.) c. Use the vendor's profit function to determine the number of gyro combos that should be prepared and sold each week to maximize profit. What is the maximum weekly profit? The maximum weekly profit is $\$ \square$ when $\square$ gyro combos are prepared and sold. (Type integers or decimals rounded to two decimal places as needed.)
Solution
Step 1 ：Let's denote the number of gyro combos prepared and sold each week as $x$. The weekly cost function, $C(x)$, for the food truck vendor is the sum of fixed and variable costs, which is $560 + 5.30x$.
Step 2 ：The revenue function, $R(x)$, is given as $-0.001x^2 + 8.54x$. The profit function, $P(x)$, is the difference between the revenue and the cost functions, which is $R(x) - C(x) = -0.001x^2 + 8.54x - (560 + 5.30x) = -0.001x^2 + 3.24x - 560$.
Step 3 ：To find the number of gyro combos that should be prepared and sold each week to maximize profit, we need to find the maximum of the profit function. The profit function is a quadratic function, and its maximum occurs at the vertex. The x-coordinate of the vertex of a quadratic function $ax^2 + bx + c$ is given by $-b/2a$. So, the number of gyro combos that should be prepared and sold each week to maximize profit is $x = -3.24/(2*(-0.001))$.
Step 4 ：Substitute $x$ into the profit function $P(x)$ to get the maximum weekly profit. The maximum weekly profit is $P(x) = -0.001*(-3.24/(2*(-0.001)))^2 + 3.24*(-3.24/(2*(-0.001))) - 560$.
Step 5 ：Finally, we need to round the results to two decimal places as needed.
