Question

Please help Find x​

Answer 1

Answer:  x = 12

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Explanation:

The hexagon is broken up into 6 congruent or identical equilateral triangles. If we find the area of one triangle, then we multiply by 6 to get the area of the hexagon.

Going in reverse, we divide the hexagon's area by 6 to get the area of one equilateral triangle

We're told the hexagon has an area of $18\sqrt{3}$ square inches. Divide this by 6 and you should get the result $3\sqrt{3}$. So each of the six equilateral triangles has area of $3\sqrt{3}$ square inches.

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Each triangle has a base of $2(\sqrt{3})^{x/12}$ and a height of $(\sqrt{3})^{x/6}$

The x/12 and x/6 are exponents.

Let $b = 2(\sqrt{3})^{x/12}$ and $h = (\sqrt{3})^{x/6}$ be the base and height respectively.

Also, let $A = 3\sqrt{3}$ be the area of the triangle

We can then solve for x like so:

$A = 0.5*b*h\\\\3\sqrt{3} = 0.5*2(\sqrt{3})^{x/12}*(\sqrt{3})^{x/6}\\\\3*3^{1/2} = (\sqrt{3})^{x/12+x/6}\\\\3^1*3^{1/2} = (\sqrt{3})^{x/12+2x/12}\\\\3^{1+1/2} = (\sqrt{3})^{3x/12}\\\\3^{3/2} = (3^{1/2})^{x/4}\\\\3^{3/2} = 3^{(1/2)*(x/4)}\\\\3^{3/2} = 3^{x/8}$

Since the bases are equal to 3, this means the exponents must be equal as well (for both sides overall to be equal)

3/2 = x/8

3*8 = 2*x ... cross multiply

24 = 2x

2x = 24

x = 24/2

x = 12