Find the dimensions of the rectangle of maximum area that can be inscribed in a right triangle with base 6 units and height 4 units?

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Find the dimensions of the rectangle of maximum area that can be inscribed in a right triangle with base 6 units and height 4 units?

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Answer 1 · 2018-03-06T23:24:09

Rectangle will be $3$ $3$ units long and have height of $2$ $2$ units.

Explanation:

Consider the area of the enclosing triangle, since it is a right triangle it's area is given by $A = \frac{B H}{2}$ $A=[BH]/2$ = $12 u n i t s^{2}$ $12units^2$ in this case.
Let the area of the rectangle = $x y$ $xy$ ............ $[1]$ $[1]$

Consider the two small triangles left over by the intrusion of the rectangle.

The one lying on the $x$ $x$ axis will have dimensions $[6 - x] \frac{y}{2}$ $[6-x]y/2$
The one on the $y$ $y$ axis will have dimensions $[4 - y] \frac{x}{2}$ $[4-y]x/2$

So the total area of the triangle = $12 = x y + [6 - x] \frac{y}{2} + [4 - y] \frac{x}{2}$ $12=xy+[6-x]y/2+[4-y]x/2$

Multiplying both sides by $2$ $2$ and expanding the brackets......

$24 = 2 x y + 6 y - x y + 4 x - x y$ $24=2xy+6y-xy+4x-xy$ . i.e .... $24 = 6 y + 4 x$ $24 =6y+4x$ and so

$y = \frac{24 - 4 x}{6}$ $y=[24-4x]/6$ and substituting this value for $y$ $y$ in....... $[1]$ $[1]$

$x y$ $xy$ = $[x] \frac{24 - 4 x}{6}$ $[x][24-4x]/6$ ..= $24 \frac{x}{6} - 4 \frac{x^{2}}{6}$ $24x/6-4x^2/6$ = $[4 x - 2 \frac{x^{2}}{3}]$ $[4x-2x^2/3]$ .

We can now differentiate this with respect to the area A. [since we have the variables in terms of of $x$ $x$ ]

$d \frac{A}{d x} = 4 - 4 \frac{x}{3}$ $dA/dx=4-4x/3$ =0 for max or min, therefore $x = 3$ $x=3$ . To see if this value of $x$ $x$ represents a maximum turning point , and thus maximises the Area in terms of $x$ $x$ we need the second derivative test.

$d^{2} \frac{A}{{d x}^{2}}$ $d^2A/[dx^2$ = $- \frac{4}{3}$ $-4/3$ which is negative whatever the value of x and so represents a maximum turning point.

Substituting this value of $x$ $x$ = $3$ $3$ in $y = \frac{24 - 4 x}{6}$ $y = [24-4x]/6$ , $y = 2$ $y =2$

Sorry I am unable to include a sketch, getting on a bit now and technology has left me behind somewhat.

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Find the dimensions of the rectangle of maximum area that can be inscribed in a right triangle with base 6 units and height 4 units?

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