Expert: Sherry Wallin - 1/22/2009

Question
two sides of a triangle are 5 and 11. What length for the third side will maximize the area of the triangle

Answer
Hi Rich~
    After careful thought and consideration I think I have a solution method that does not require calculus. Here it is:

The largest area of any triangle (with the same perimeter) is an equilateral triangle. We already have sides that are not identical so we know we won't have an equilateral triangle and cannot get the absolute maximum.

There is Heron's formula for calculating the area of any triangle:
sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2 (the semi perimeter of the triangle), and a,b,c are triangle side lengths. In our case let a = 5 and b = 11 and c = x.

s = [(5+11+x)/2] = (16+x)/2  and A is the area of the triangle
A = sqrt{[(16+x)/2]([(16+x)/2]-5)([(16+x)/2]-11)([(16+x)/2]-x)}

get common denominators:
A = sqrt{[(16+x)/2]([(16+x-10)/2]([(16+x-22)/2]([(16+x-2x)/2]}

simplify
A = sqrt{[(16+x)/2]([(x+6)/2]([(x-6)/2]([(16-x)/2]}

change sign or factor out a -1 from 16-x, also rewrite 16+x as x+16
A = sqrt{-1[(x+16)/2]([(x+6)/2]([(x-6)/2]([(x-16)/2]}

we have 4 factors of 2 in the denominator, 2^4 = 16 so rewrite and rearrange order of linear factors moving (x-16) next to (x+16):
sqrt{(-1/16)(x+16)((x-16)((x+6)((x-6)}

square both sides to get rid of the sqrt resulting in
A^2 = (-1/16)(x+16)((x-16)((x+6)((x-6)

multiply linear factors:
A^2 = (-1/16)(x^2-256)(x^2-36)

let u = x^2  and multiply resulting in:
A^2 = (-1/16)(u-256)(u-36)= (-1/16)(u^2 - 292u + 9216)

At this point there are a number of ways to factor the quadratic, I will use completing the square:
A^2 = (-1/16)(u^2 - 292u + (-146)^2 - (-146)^2 + 9216)
A^2 = (-1/16)[(u -146)^2 - (-146)^2 + 9216]
A^2 = (-1/16)[(u -146)^2 - 21316 + 9216]
A^2= (-1/16)[(u -146)^2 - 12100]

multiply through by (-1/16) getting:
A^2 = (-1/16)[(u -146)^2]-(-1/16)(12100)
   = (-1/16)[(u -146)^2]+(1/16)(12100)
   = (-1/16)[(u -146)^2] + 12100/16
   = (-1/16)[(u -146)^2] + 756.25

Now we have the area squared and the first term will be negative regardless of the value of u. In order for the expression on the right to be the largest we must have (-1/16)[(u -146)^2] = 0 and that will only happen if u = 146. Now recall u = x^2 so take the positive square root of 146 (we don't need the negative value since measurement is always positive) to find that value of x that will give us the largest area. So x is 12.08 rounded to the nearest hundredth. This is the length of the 3rd side that will give you the largest area.

Don't worry about the fact that we squared the area originally because we were last looking at A^2 = (u-146)^2 + some number [by the way that some number is the maximum area] but take the sqrt of both sides and you have
A = sqrt[(u-46)^2 + some number]> sqrt(u-146)^2 = u-146. Another way to think about this is if you have the maximum area squared you will still have the maximum area if you take the sqrt of the squared area. I hope this makes sense.

I believe in my last reply I told you, you could check all whole number values between 6 and 16, i.e., check 7,8,9,10,11,12,13,14,15 and I also said 12 would give you the largest value. I said the reason one would only have to check these whole number values was because of the triangle inequality that says the sum of any two sides must be larger than the 3rd side. 5+11 = 16 and 16 > 12.08, so our answer seems reasonable also. But if you needed to show how to find the largest area and not just a whole number we have derived the way to determine it to be actually 12.08.

Good luck, I hope this answer helps you in your further studies.

Math Prof