Linear programming word question

[Badatmath_]

Question: A florist can make a grand arrangement in 25 minutes or a simple arrangement in 10 minutes. The florist makes at least twice as many of the simple arrangement as the grand arrangements, and he makes at least one of each type of arrangement every day. If the florist works part time for 3 hours per day, and the profit on the simple arrangements is $15 and the profit on the grand arrangements is $20, find the number and type of arrangements that the florist should produce to maximize profit. What is the maximum profit?

(Let x = Number of grand arrangements and Let y = Number of small arrangements
I have created the time limitation function of 25x+10y \le 180 and the profit function of 30x+15y=p (profit) as well as set my limitations for x and y (x /ge 1 and y /ge 1) but I don't know what to do now. If someone could solve this as well as provide the two functions that intersect at the max point that'd be great.

 

[MarkFL]

Yes, the objective function, that which we are seeking to optimize, is the profit function:

[MATH]P(x,y)=20x+15y[/MATH]
Subject to the constraints:

[MATH]y\le2x[/MATH]
[MATH]25x+10y\le180[/MATH]
[MATH]1\le x[/MATH]
[MATH]1\le y[/MATH]
Graphing the region that satisfies all the constraints, we get:



Can you proceed?

 

[Badatmath_]

Yes, the objective function, that which we are seeking to optimize, is the profit function:

[MATH]P(x,y)=20x+15y[/MATH]
Subject to the constraints:

[MATH]y\le2x[/MATH]
[MATH]25x+10y\le180[/MATH]
[MATH]1\le x[/MATH]
[MATH]1\le y[/MATH]
Graphing the region that satisfies all the constraints, we get:

View attachment 11689

Can you proceed?

Well seeing as the vertice (4,8) satisfies the twice as many requirement and produces the largest profit number, I'd assume that it is the answer.

 

[MarkFL]

Well seeing as the vertice (4,8) satisfies the twice as many requirement and produces the largest profit number, I'd assume that it is the answer.

I would agree.