How can I calculate the (A) area of the triangle as the function of x

Question

A right triangle has one vertex on the graph of y=x³ , x>0, at (x,y), another at the origin, and the third on the positive y-axis at (0, y). Express the (A) area of the triangle as a function of x  

Answer 1

Logan--

 

It helps if you can plot the curve yourself.  

 

A triangle has 3 vertices and the area of a triangle can be found by the formula

Area = 1/2*base*height    or   A = 1/2*b*h.

 

 

One vertex is fixed at the origin, at (0,0).  The second is at (0,y), and the third at (x,y) which lies on the curve y=x^3.  Therefore, we can rewrite point 3 as (x,y)=(x,x^3).  Similarly, point two can be written as (0,y)=(0,x^3).

 

So, all that is left is to find the length of the horizontal side (or, the "base") and vertical side (the "height") so we can plug into the formula.

 

To find the length of the base, note that we move from (0,0) to (x,x^3), which is x units horizontally.  Similarly, for the height, we move from (0,0) to (0,x^3) which is x^3 units vertically.

 

Plugging into our formula:

A = 1/2*b*h

A = 1/2*x*x^3

A = 1/2*x^4,

 

 

and so, we find that the area is a function of x.

Answer 2

Area of the triangle is.  =0.5*x^4

 

Surendra

 

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