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 = ![\pi x \frac{\sqrt{a} }{b}](https://tex.z-dn.net/?f=%5Cpi%20x%20%5Cfrac%7B%5Csqrt%7Ba%7D%20%7D%7Bb%7D)
Let X be the side of the hexagon.
So, the area of the Hexagon is:
Area of the Hexagon = ![\frac{3\sqrt{3} }{2} X^{2}](https://tex.z-dn.net/?f=%5Cfrac%7B3%5Csqrt%7B3%7D%20%7D%7B2%7D%20%20X%5E%7B2%7D)
Now, we have to calculate the length of Fido's Leash by using Lw of Sine
Law of Sine =
=
= ![\frac{c}{sinC}](https://tex.z-dn.net/?f=%5Cfrac%7Bc%7D%7BsinC%7D)
= ![\frac{Length}{sin60}](https://tex.z-dn.net/?f=%5Cfrac%7BLength%7D%7Bsin60%7D)
Solving for Length, where sin30 = 1/2 and sin60 = ![\frac{\sqrt{3} }{2}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt%7B3%7D%20%7D%7B2%7D)
Suppose Length = Y
Y =
X
Now, as we have got the length we can calculate the area that Fido can cover,
So, the area covered by Fido =
x ![Y^{2}](https://tex.z-dn.net/?f=Y%5E%7B2%7D)
Area =
x
Area =
x
x ![X^{2}](https://tex.z-dn.net/?f=X%5E%7B2%7D)
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 = ![\pi x \frac{\sqrt{a} }{b}](https://tex.z-dn.net/?f=%5Cpi%20x%20%5Cfrac%7B%5Csqrt%7Ba%7D%20%7D%7Bb%7D)
x
x
/
x
/
6
/ ![12\sqrt{3}](https://tex.z-dn.net/?f=12%5Csqrt%7B3%7D)
/ ![2\sqrt{3}](https://tex.z-dn.net/?f=2%5Csqrt%7B3%7D)
Which can be rewritten as by multiplying and dividing by ![\sqrt{3}](https://tex.z-dn.net/?f=%5Csqrt%7B3%7D)
/ 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