Current Attempt in Progress You run a small furniture business. You sign a deal with a customer to deliver up to 400 chairs, the exact number to be determined by the customer later. The price will be $\$ 100$ per chair up to 300 chairs, and above 300 , the price will be reduced by $\$ 0.25$ per chair (on the whole order) for every additional chair over 300 ordered. What is the largest revenue your company can make under this deal? \[ R=\$ \quad \mathbf{i} \] eTextbook and Media What is the smallest revenue your company can make under this deal? \[ R=\$ \] i eTextbook and Media

Step 1 ：Let R be the revenue and n be the number of chairs ordered. We can write a function to calculate the revenue based on the number of chairs:

Step 2 ：\[ R(n) = \begin{cases} 100n & \text{if } n \leq 300 \\ 100n - 0.25(n-300)^2 & \text{if } n > 300 \end{cases} \]

Step 3 ：To find the largest revenue, we need to find the maximum value of R(n) for n > 300. We can do this by taking the derivative of R(n) with respect to n and setting it to 0:

Step 4 ：\[ \frac{dR}{dn} = 100 - 0.5(n-300) \]

Step 5 ：\[ 0 = 100 - 0.5(n-300) \]

Step 6 ：\[ n = 400 \]

Step 7 ：Since the maximum value occurs at n = 400, we can plug this value back into the revenue function to find the largest revenue:

Step 8 ：\[ R(400) = 100(400) - 0.25(400-300)^2 \]

Step 9 ：\[ R(400) = \boxed{30,625} \]

Step 10 ：To find the smallest revenue, we need to consider the case when the customer orders the minimum number of chairs, which is 1:

Step 11 ：\[ R(1) = 100(1) \]

Step 12 ：\[ R(1) = \boxed{100} \]

Step 13 ：The largest revenue the company can make under this deal is $30,625, and the smallest revenue is $100.