The sum of the base and the height of a triangle is 20cm Find the demesions for which the area is maximum?

Answer 1

If we say that the base width is "b", the height is "h", and the area is "a", then we are told that:

b + h = 20

∴h = 20 - b

Now, given the nature of a triangle, we know that it's area is half it's base times it's height, so we can also say:

a = bh/2

Now we can combine those equations, narrowing it down to one variable

∴a = b(20 - b)/2

∴a = 10b - b2/2

This equation gives us a parabola. Looking at the equation, you can see that it's an inverted parabola, with it's peak pointing "upwards". To find the point at which that parabola peaks, we need to know when it has a slope of 0. To find that point, we first need the slope of the curve, which we can get by taking it's derivative:

a = 10b - b2/2

∴a' = 10 - b

if a' = 0, then b = 10.

∴ The highest point on that curve occurs when b = 10, ie, when it's base has a width of ten and when it's height is also 10. This makes sense. Consider: the highest ratio of area to border in a rectangle will happen when that rectangle is a square. Since the area of a rectangle is in fact half the area of the a square with the same width and height, that same relationship will hold true. So in order to get the maximum ratio of area to perimeter in a triangle, it's width and height should be the same.