If the perimeter of an isoceles triangle is 18cm, determine the maximum area of he triangle using calculus methods

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should remember that an isosceles triangle has two equal sides, hence you may evaluate the perimeter such that:

`P = x + x + y`

`18 = 2x + y =gt y = 18 - 2x`

You should evaluate the area of triangle such that:

`A = (x*y*sin alpha)/2`

You need to substitute `18 - 2x`  for y in equation of area such that:

`A(x) = x*(18-2x)*(sin alpha)/2`

`A(x) = 9x sin alpha - x^2*sin alpha`

`A(x) = sin alpha(9x - x^2)`

You need to determine the maximum area of triangle, hence you need to solve for x the equation A'(x) = 0 such that:

`A'(x) = sin alpha*(9 - 2x)`

You need to solve the equation `sin alpha*(9 - 2x) = 0`  such that:

`sin alpha = 0`  => a lpha = 0

`9 - 2x = 0 =gt 2x = 9 =gt x = 9/2 =gt x = 4.5 cm`

`y = 18 - 9 =gt y = 9 cm`

Hence, the area of triangle is maximum at x = 4.5 cm and y = 9 cm.

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