Question

What is the maximum possible area of all right triangles whose sum of two right sides is 12?​

Answer 1

To find the maximum possible area of all right triangles whose sum of two right sides is 12, we can use the formula for the area of a right triangle, which is 1/2 * base * height.

In this case, let's assume one of the right sides is x, which means the other right side would be 12 - x.

To maximize the area, we want to maximize the product of the two sides, which is x * (12 - x).

To find the maximum value of this product, we can use a concept called optimization. We can take the derivative of the product with respect to x, set it equal to zero, and solve for x.

After doing the math, we find that the maximum value of the product occurs when x = 6.

Plugging this value back into the formula for the area, we get:
Area = 1/2 * x * (12 - x)
= 1/2 * 6 * (12 - 6)
= 1/2 * 6 * 6
= 18

So, the maximum possible area of all right triangles with a sum of two right sides equal to 12 is 18.

I hope that helps! Let me know if you have any other questions.

Answer 2

Answer: 18 square units

Work Shown

x = horizontal leg = base

12-x = vertical leg = height

area = 0.5*base*height

area = 0.5*x*(12-x)

area = -0.5x^2 + 6x

Use a graphing calculator to find that (6,18) is the highest point on this parabola. The parabola opens downward because the leading coefficient -0.5 is negative.

Another way to find the highest point is to find the roots of y = -0.5x^2+6x, and then find the midpoint of the roots. This will determine the axis of symmetry. Plug this value into y = -0.5x^2+6x so you can find the y value.

(6,18) being the highest point means x = 6 leads to maxArea = 18.

If x = 6 is the base, then 12-x = 12-6 = 6 is the height. This right triangle is isosceles.