3 A florist shop plans to sell roses by the dozen for Valentine's Day. When they set their price at \$25 per dozen, they can sell 150 dozen, but if they raise the price to $\$ 35$ per dozen, sales drop to 100 dozen. Assuming demand for roses depends linearly on price, what price per dozen gives the maximum revenue, and what does that revenue amount to?

Step 1 ：Given two points on the demand curve: (25, 150) and (35, 100), we can find the equation of the line passing through these points, which will give us the demand as a function of price.

Step 2 ：Then we can find the revenue as a function of price by multiplying the price by the demand.

Step 3 ：The maximum revenue will occur at the vertex of this parabola, which we can find by completing the square or by using the formula for the vertex of a parabola.

Step 4 ：By solving the equation, we find that the critical point is at \(p = 27.5\).

Step 5 ：Substituting \(p = 27.5\) into the revenue equation, we find that the maximum revenue is \(3781.25\).

Step 6 ：Final Answer: The maximum revenue is achieved when the price per dozen is \(\boxed{27.50}\) dollars, and the maximum revenue is \(\boxed{3781.25}\) dollars.