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Fido's leash is tied to a stake at the center of his yard, which is in the shape of a regular hexagon. His leash is exactly long enough to reach the midpoint of each side of his yard. If the fraction of the area of Fido's yard that he is able to reach while on his leash is expressed in simplest radical form as [(sqrt{a})/b] * pi, what is the value of the product ab?

Sagot :

Answer:

a = 3

b = 6

The product ab =  3 x 6 = 18

Step-by-step explanation:

Data Given:

Fraction of the area of Fido's yard = [tex]\pi x \frac{\sqrt{a} }{b}[/tex]

Let X be the side of the hexagon.

So, the area of the Hexagon is:

Area of the Hexagon = [tex]\frac{3\sqrt{3} }{2} X^{2}[/tex]

Now, we have to calculate the length of Fido's Leash by using Lw of Sine

Law of Sine = [tex]\frac{a}{sinA}[/tex] = [tex]\frac{b}{sinB}[/tex] = [tex]\frac{c}{sinC}[/tex]

[tex]\frac{0.5S}{sin30}[/tex] = [tex]\frac{Length}{sin60}[/tex]

Solving for Length, where sin30 = 1/2 and sin60 = [tex]\frac{\sqrt{3} }{2}[/tex]

Suppose Length = Y

Y = [tex]\frac{\sqrt{3} }{2}[/tex]X

Now, as we have got the length we can calculate the area that Fido can cover,

So, the area covered by Fido = [tex]\pi[/tex] x [tex]Y^{2}[/tex]

Area = [tex]\pi[/tex] x [tex](\frac{\sqrt{3} }{2}X) ^{2}[/tex]

Area = [tex]\pi[/tex] x [tex]\frac{3}{4}[/tex] x [tex]X^{2}[/tex]

So, now, we can get the fraction of the area of Fido's Yard by dividing the area of hexagon by area that fido can cover.

Fraction of the area of Fido's yard = [tex]\pi x \frac{\sqrt{a} }{b}[/tex]

[tex]\pi[/tex] x [tex]\frac{3}{4}[/tex] x [tex]X^{2}[/tex] / [tex]\frac{3\sqrt{3} }{2} X^{2}[/tex]

[tex]\pi[/tex] x [tex]\frac{3}{4}[/tex] / [tex]\frac{3\sqrt{3} }{2}[/tex]

6 [tex]\pi[/tex] / [tex]12\sqrt{3}[/tex]

[tex]\pi[/tex] / [tex]2\sqrt{3}[/tex]

Which can be rewritten as by multiplying and dividing by [tex]\sqrt{3}[/tex]

[tex]\sqrt{3} \pi[/tex]/ 6

By comparing the above equation and the given equation we can find a and b :

So,

a = 3

b = 6

And the product ab =  3 x 6 = 18

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