The sum of the base and the height of a triangle is 30 cm. find the dimensions for which the area is a maximum. the triangle with the maximum area has a height of and a base of

If the sum of the base and the height are 30, then b+h=30 or h=b-30.

Remember that the area of any triangle is A=½bh.

If you substitute in b-30 for h, then you create a function for the area in terms of the base.

A = ½b(30-b)=15b-½b2

The maximum area would be at the vertex of the parabola that would result from the function.  We can graph to find the max, but there is a simple formula for the x-coordinate of the vertex: -b/2a where b=the linear coefficient and a=the quadratic coefficient.

-15/2(-½)=15  meaning that a base of 15cm would provide the maximum area.  The corresponding height would be 30-15 or 15cm as well.

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