Question

The dwarf queen wants to build a tunnel between her cities Yram and Haras.
To plan the tunnel's length, the queen first walks 7.0 km from Yram to a point where she can see both
cities. From that point, she measures 29° between the cities, as shown. Lastly, she walks 6.0 km to Haras.
Yram
7 km
29
Tunnel?
6 km
Haras
What is the length of the tunnel?
Do not round during your calculations. Round your final answer to the nearest tenth of a kilometer.
km

Answer 1

7/10 is the correct answer

Answer 2

The length of the tunnel calculated using the cosine rule of triangles is approximately 3.4 km.

Let us consider a triangle ABC such that vertex A denotes Haras, vertex C represents Yram and $\angle ABC = 29^{\circ} , BA=7 ,BC=6$  . The detailed diagram of the triangle is attached below.

Now in triangle ABC, we will use the Cosine rule to calculate the unknown length of the side of the triangle.

The cosine rule states that in any triangle where the side lengths are represented by a,b and c

${\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma }$

Now in the given triangle, we have the sides whose lengths are given and the included angle is also given. Using these values we calculate:

${\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma }\\\\or, c^2= 7^2+6^2-2\times 7 \times 6 \times \cos 29^{\circ}\\\\or, c^2 = 49+36 -73.468055\\\\or, c^2 = 11.53194..\\\\or, c = 3.3958...\\\\or, c\approx 3.4$

Therefore the length of the unknown side of the triangle is approximately 3.4 km calculated by the law of cosines.

To learn more about law of cosines visit:

https://brainly.in/question/13440851

https://brainly.in/question/10457906

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