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The sides of a right-angled triangle are x and 8 - 4x. Find the dimensions of the triangle if its area is a maximum.

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Final answer:

To find the dimensions of the triangle with maximum area, use the area formula A = (1/2) * base * height. Take the derivative, set it equal to zero, and solve for x. Discard any invalid solutions.

Step-by-step explanation:

To find the dimensions of the right-angled triangle with maximum area, we need to consider the relationship between the sides of the triangle and its area. The area of a triangle can be calculated using the formula A = (1/2) * base * height. In this case, the base is x and the height is 8 - 4x. So, the formula for the area is A = (1/2) * x * (8 - 4x).

To maximize the area, we can take the derivative of the area formula with respect to x and set it equal to zero. This will give us the critical point where the area is maximized. Taking the derivative and simplifying, we get the equation -6x^2 + 8x = 0. Solving for x, we find x = 0 or x = 4/3.

We can discard the solution x = 0 because it would make the triangle degenerate to a line. So, the dimensions of the triangle with maximum area are x = 4/3 and 8 - 4x = 8 - 4(4/3) = 8 - 16/3 = 8/3.

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