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A right triangle has one vertex on the graph of $y=x_{3}$ at the point $(x,y)$ in the first quadrant, another at the origin, and the third on the positive $y$-axis at the point $(0,y)$, as shown in the figure below. a) Express $A$, the area of the triangle, as a function of $x$. State the natural domain of $A=A(x)$. b) What is the area of this triangle when $x=4$ ? c) Explain why this area cannot be maximized.

(a) We know that, Area of right triangle =

So,

Area of the given right triangle = Since,

So, Area of the given right triangle = A(x) =

Therefore, Area of the given right triangle is found in terms of x.

Now, we have to determine the natural domain of A = A(x) .

Since, we can put any value the above function.

So, the natural domain of A = A(x) is:

The area and natural domain is calculated using required formula and concepts.

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