Optimization | box problem

[elisaflemer]

Hi! I was doing this problem but couldn't figure out the interval for x (in pink). I imagined that the bigger it got, the smaller y would get, so I rounded it to 0 and solved the equation for x. I got a 55.1228, but I'm not sure that's right. I ended up solving the problem with the second derivative test, but I would like to clear up this interval thing.
Could anyone help me out, please?

QUESTION: A printer needs to make a poster that will have a total area of 200 in2 and will have 1 inch margins on the sides, a 2 inch margin on the top and a 1.5 inch margin on the bottom as shown below. What dimensions will give the largest printed area?

Answer: width 10.6904, length: 18.7084
My resolution (I probably made this harder than it should be to be honest):

 

[Deleted member 4993]

Hi! I was doing this problem but couldn't figure out the interval for x (in pink). I imagined that the bigger it got, the smaller y would get, so I rounded it to 0 and solved the equation for x. I got a 55.1228, but I'm not sure that's right. I ended up solving the problem with the second derivative test, but I would like to clear up this interval thing.
Could anyone help me out, please?

QUESTION: A printer needs to make a poster that will have a total area of 200 in2 and will have 1 inch margins on the sides, a 2 inch margin on the top and a 1.5 inch margin on the bottom as shown below. What dimensions will give the largest printed area?

Answer: width 10.6904, length: 18.7084
My resolution (I probably made this harder than it should be to be honest):
View attachment 26057

The minimum height of the poster is 3.5" \(\displaystyle \ \to \ \)max x = (200/3.5 - 2)"