Question

The diagram shows a hexagon-shaped tile used for flooring. Each hexagon tile has an area of 18/3 in 2 Find x.​

Answer 1

Answer:

x = 12

Step-by-step explanation:

The hexagonal tile given here is made up of 6 triangles with equal bases and heights.

So, area of hexagonal tile will be equal to 6 times the area of one triangle.

Therefore,

$18 \sqrt{3} = 6 \times \frac{1}{2} \times 2 {( \sqrt{3)} }^{ \frac{x}{12} } . {( \sqrt{3)} }^{ \frac{x}{6} } \\ \\ 18 \sqrt{3} = 6 {( \sqrt{3)} }^{ \frac{x}{12} } . {( \sqrt{3)} }^{ \frac{2x}{12} } \\ \\ \frac{18 \sqrt{3} }{6} = {( \sqrt{3)} }^{ \frac{x}{12} + \frac{2x}{12} } \\ \\ 3 \sqrt{3} = {( \sqrt{3)} }^{ \frac{3x}{12} } \\ \\ 3. {3}^{ \frac{1}{2} } = {3}^{ \frac{3x}{24} } \\ \\ {3}^{1 + \frac{1}{2} } = {3}^{ \frac{x}{8} } \\ \\ {3}^{ \frac{3}{2} } = {3}^{ \frac{x}{8} } \\ (bases \: are \: equal \: so \: exponents \: \\ will \: also \: be \: equal) \\ \implies \frac{3}{2} = \frac{x}{8} \\ \\ x = \frac{3 \times 8}{2} \\ \\ x = 3 \times 4 \\ \\ x = 12$

Answer 2

The area of the hexagon-shaped tile is the amount of space it occupies

The value of x is 2

The given parameters are:

$\mathbf{Area = 18\sqrt 3}$

$\mathbf{Base= (\sqrt 3)^{\frac x6}}$

$\mathbf{Height = 2(\sqrt 3)^{\frac x{12}}}$

The shape is a regular hexagon (i.e. it has 6 equal sides).

So, the area of one of the segments is the total area divided by 6 and this is calculated as:

$\mathbf{A(1) = \frac{18\sqrt 3}6}$

$\mathbf{A(1) = 3\sqrt 3}$

A segment of the hexagon is a triangle, and the area of a triangle is:

$\mathbf{Area = \frac 12 \times Base \times Height}$

So, we have:

$\mathbf{3\sqrt 3= \frac 12 \times (\sqrt 3)^{\frac x6} \times 2(\sqrt 3)^{\frac{x}{12}}}$

This gives

$\mathbf{3\sqrt 3= (\sqrt 3)^{\frac x6} \times \sqrt 3)^{\frac{x}{12}}}$

Apply law of indices

$\mathbf{3\sqrt 3= (\sqrt 3)^{\frac x6 + \frac{x}{12}}}$

Add the exponents

$\mathbf{3\sqrt 3= (\sqrt 3)^{\frac x4}}$

Express the equations as a base of 3

$\mathbf{3^{\frac 32}= 3^{\frac x8}}$

Cancel out the base

$\mathbf{\frac 32= \frac x8}$

Multiply through by 8

$\mathbf{12= x}$

Rewrite as:

$\mathbf{x = 12}$

Hence, the value of x is 2

Read more about areas at:

https://brainly.com/question/3365788