Can you help with this problems????

**Susie38** · May 29th 2006, 01:47 PM

Let T be a triangle witht vertices (0,0), (0,c^2), (c,c^2) and let R be the region between y =cx and y = x^2 where c>0. Fidn the ratio area T/ area R.

**Soroban** · May 29th 2006, 05:40 PM

Hello, Susie38!

It's a straight-forward area problem.
Exactly where does your difficulty lie?

Let $T$ be a triangle with vertices $(0,0),\;(0,c^2),\;(c,c^2)$
and let $R$ be the region between $y = cx$ and $y = x^2,\; c > 0$
Find the ratio $\frac{area\:T}{area\:R}$

Code:


            |
      (0,c²)* - - - - - * (c,c²)
            |         /:
            |       /::*
            |     /::::
            |   /::::*
            | /:::*
      - - - **- - - - - -
          (0,0)

The base of the triangle is $c$ , its height is $c^2$ .
. . Its area is: $A_T \:=\:\frac{1}{2}(c)(c^2) \:=\:\frac{1}{2}c^3$

The area of the parabolic segment is:
. . $\displaystyle{A_R \;= \;\int^c_0(cx - x^2)\ dx \;=\;\frac{c}{2}x^2 - \frac{1}{3}x^3\bigg|^c_0 \;= \;\frac{1}{6}c^3$

Therefore: $\displaystyle{\frac{A_T}{A_R}\;=\;\frac{\frac{1}{2 }c^3}{\frac{1}{6}c^3} \;= \;3$

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