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. $\displaystyle s_1$ means a side of the first garden...

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

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

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

Then: $\displaystyle s_2=12.5-s_1$

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

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

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

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

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

Subtract: $\displaystyle 25s_1-156.25\leq100$

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

Divide by 25: $\displaystyle s_1\leq10.25$

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