1. ## Optimizing fence

Thank You in advance to those who try and solve this problem

Chris needs 50m of fencing to enclose two separate square gardens whose area must differ by no more than 100m squared. What is the greatest length that a side of the larger square garden can have.

2. Originally Posted by begold91
Thank You in advance to those who try and solve this problem

Chris needs 50m of fencing to enclose two separate square gardens whose area must differ by no more than 100m squared. What is the greatest length that a side of the larger square garden can have.
Alright, the large garden is garden 1. The small one is garden 2. $s_1$ means a side of the first garden...

You know that: $s_1^2-s_2^2\leq100$

And that: $4s_1+4s_2=50$

Thus: $s_1+s_2=\frac{50}{4}=12.5$

Then: $s_2=12.5-s_1$

Now go back to this one: $s_1^2-s_2^2\leq100$

Substitute: $s_1^2-\left(12.5-s_1\right)^2\leq100$

Then: $s_1^2-\left(156.25-25s_1+s_1^2\right)\leq100$

Now spread out the negative sign: $s_1^2-156.25+25s_1-s_1^2\leq100$

Group like terms: $s_1^2-s_1^2+25s_1-156.25\leq100$

Subtract: $25s_1-156.25\leq100$

Add 156.25 to both sides: $25s_1\leq256.25$

Divide by 25: $s_1\leq10.25$

So the greatest length $s_1$ can have is 10.25. But check my arithmetic!!!