The sum of the base and height of a triangle is 28cm. Find the dimensions for which the area is the maximum.

Question

The sum of the base and height of a triangle is 28cm. Find the dimensions for which the area is the maximum.

Answer 1

Just take some guesses at the b and h. Area is 1/2 times b times h, so check the product of b *h by trial and error (How's that for being "mathy"?)

1,27= 27 
2, 26= 52 
3, 25= 75 
4, 24 
5, 23 
6, 22 =132
7, 21 
8, 20 
9, 19 
10, 18 = 180

11, 17 = 187

12, 16 = 192

13, 15 = 195
14, 14 = 196

I actually didn't have to check so many. I see the pattern. Therefore, I know that the dimensions 14 and 14 will give the largest area. It doesn't matter which is the height and which is the base.

 

Without using calculus, here's another method which uses the graph and properties of a parabola which will face downward.

 

b + h = 28 
b = 28 - h 

We must find where the maximum "is" of the function that can be graphed as a parabola.
Area = 1/2 * b * h = 1/2 *(h(28-h))

The 1/2 can be ignored when finding zeros (setting function = 0) and find the zeros of h(28-h).

It faces down and the zeros are at h=28 and h=0. The maximum which is at the vertex is found half way between the zeros at (28+0)/2 = 14 =h = b ---same answer.