Problem
(1 point) A restaurant chain is considering raising the prices on their burgers. They have found that for every $\$ 1$ increase, their customers will buy 50 fewer burgers each day. Say that the price of a burger is currently $\$ 4$, and their customers buy 850 burgers per day. a) Write a formula $p(x)$, which gives the price per burger (in dollars) after an increase of $x$ dollars per burger. \[ p(x)=\square \] b) Write a formula $k(x)$, which represents the number of burgers the chain sells each day after an increase of $x$ dollars per burger. \[ k(x)=\square \] c) Which of the following gives an interpretation of $p(x) \cdot k(x)$ in real-world terms. A. $p(x) \cdot k(x)$ gives the daily revenue of the restaurant 4 ter an increase of $x$ dollars per burger. B. $p(x) \cdot k(x)$ gives the daily profit of the restaurant after an increase of $x$ dollars per burger. C. $p(x) \cdot k(x)$ gives the increase in profit after an increase of $x$ dollars per burger. D. $p(x) \cdot k(x)$ gives the total price of a burger after an increase of $x$ dollars per burger. d) Find the approximate value of $x$ which maximizes $p(x) \cdot k(x)$. \[ x=\square \text { dollars } \] Preview My Answers Submit Answers
Solution
Step 1 ：a) The formula for the price per burger after an increase of $x$ dollars per burger is \(p(x) = 4 + x\).
Step 2 ：b) The formula for the number of burgers sold each day after an increase of $x$ dollars per burger is \(k(x) = 850 - 50x\).
Step 3 ：c) The interpretation of \(p(x) \cdot k(x)\) in real-world terms is that it gives the daily revenue of the restaurant after an increase of $x$ dollars per burger.
Step 4 ：d) The approximate value of $x$ which maximizes \(p(x) \cdot k(x)\) is \(\boxed{\frac{13}{2}}\) dollars.
