nickjsd
contestada

maria is planning a triangular garden she wants to build a fence around the garden to keep out the rabbits. the length of one side of the garden is 29 feet. if the angles of this side are 65 and 44 find the length of the fence needed to enclose the garden

Respuesta :

She would need 78.1 feet.

We first find the measure of the missing angle:

180-65-44 = 71

Now we use the law of sines:

sin A/a = sin B/b = sin C/c

With our information we have:
sin71/29 = sin65/a

Cross multiply:
a sin 71= 29 sin 65

Divide both sides by sin 71:
(a sin 71)/(sin 71) = (29 sin 65)/(sin 71)
a = 27.8

Using it again:

(sin 71)/29 = (sin 44)/b

Cross multiply:

b sin 71 = 29 sin 44

Divide both sides by sin 71:

(b sin 71)/(sin 71) = (29 sin 44)/(sin 71)
b = 21.3

Adding the sides:

21.3 + 27.8 + 29 = 78.1
cerverusdante
To find the angle opposite from the side of our triangle, we are going to take advantage of the fact that the sum of the interior angles of a triangle is always 180°:
[tex]A+65+44=180[/tex]
[tex]A+109=180[/tex]
[tex]A=180-109[/tex]
[tex]A=71[/tex]

To find the other two sides of our triangle, we are going to use the law of sines:
[tex] \frac{a}{sinA} = \frac{b}{sinB} [/tex]
[tex] \frac{29}{sine(71)} = \frac{b}{sine(65)} [/tex]
[tex]b= \frac{29sine(65)}{sine(71)} [/tex]
[tex]b=27.8[/tex]

[tex] \frac{a}{sineA} = \frac{c}{sineC} [/tex]
[tex] \frac{29}{sine(71)} = \frac{c}{sine(44)} [/tex]
[tex]c= \frac{29sine(44)}{sine(71)} [/tex]
[tex]c=21.3[/tex]

Now, we have all the sides of our triangle. So, the only thing left to find the length of the fence needed to enclose the garden, is add them:
[tex]P=a+b+c[/tex]
[tex]P=29+27.8+21.3[/tex]
[tex]P=78.1[/tex]

We can conclude that the length of the fence needed to enclose the garden is 78.1 feet.

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