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Sagot :
Based on the drawing obtained from the description, the location of
Manu's farm is located relatively south to Dogbe's farm.
- The distance between the two farms is approximately 3.8 km
- The bearing of Manu's farm from Dogbe is approximately 238°
Reasons:
The bearing of the cottage to Dogbe's farm = 200°
The distance Dogbe walks from the cottage to his farm = 5 km
The bearing of the cottage from Manu's farm = 110°
The distance Manu walks from the cottage to his farm = 3 km
Required:
The distance and between the two farms.
Solution:
Please find attached a drawing showing the position of the cottage
By cosine rule, we have;
- a² = b² + c² - 2·b·c·cos(A)
Where;
b = The distance Dogbe walks from the cottage to his farm = 5 km
c = The distance Manu walks to his farm from the cottage = 3 km
a = The distance between the two farms = d
A = The angle between the paths to the cottage from the farms = 50°
By plugging in the values, we have;
d² = 5² + 3² - 2 × 5 × 3 × cos(50°)
d = √(5² + 3² - 2 × 5 × 3 × cos(50°)) ≈ 3.8
- The distance between the two farms, d ≈ 3.8 km
Required:
The bearing of Manu's farm from Dogbe's
Solution:
By sine rule, we have;
- [tex]\displaystyle \frac{3}{sin(C)} = \mathbf{ \frac{d}{sin(50^{\circ})}}[/tex]
Which gives;
[tex]\displaystyle sin(C) = \mathbf{\frac{3 \times sin(50^{\circ})}{\sqrt{5^2 + 3^2 - 2 \times 5 \times 3 \times cos(50^{\circ})} }}[/tex]
[tex]\displaystyle \angle C = arcsine \left(\frac{3 \times sin(50^{\circ})}{\sqrt{5^2 + 3^2 - 2 \times 5 \times 3 \times cos(50^{\circ})} } \right) \approx 38 ^{\circ}[/tex]
The bearing of Manu's farm from Dogbe's = 200° + ∠C
Therefore;
- The bearing of Manu's farm from Dogbe's ≈ 200° + 38 =  238°
Learn more about bearings in mathematics here:
https://brainly.com/question/10710413
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