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Question

The sum of the base and the height of a triangle is 56 inches. Find the base and height for which the area is a maximum. Also, find the maximum area.

Thanks in advance... I don't even know where to start on this one.

Answer 1

Let b be the base, h be the height.

You know b+h=56.

The area is A=bh/2. Since b+h=56, that means h=56-b, so substitute this into the formula for the area. You get A=b(56-b)/2=(56b-b^2)/2.

Differentiate this, and you get 28-b. Set this equal to 0, and you get b=28.

Therefore, the height is 56-28=28. You conclude that the maximum area is 1/2*28*28=392 sq in.

Answer 2

This is a quadratic function maximum problem.

If base + height = 56 use b + h = 56. This gives h = 56 - b

Area of a triangle is 0.5bh. and then substitute in h = 56 - b.

Area = 0.5b(56 - b)
= 28b - 0.5b^2
Now complete the square on this.

Area = -0.5(b^2 - 56b)
= -0.5[(b^2 - 56b + 784) - 784]
= -0.5(b - 28)^2 + 392
Therefore the maximum area of 392 square inches occurs when b = 28. This gives h = 56 - 28 = 28.
For max area the base should be 28 inches and so should the height!

There are other ways as well...such as finding the roots of the quadratic equation, finding the midpoint to give the b coordinate of the vertex and then substituting in to find the h coordinate. Or you could always graph!

Answer 3

the maximum area is 392 inches squared...this occurs when the base and height are both 28 inches long.