A plane flies from A and B,a distance of.a 120km on a bearing of 035°. If then changes direction and flies to C,a further distance of 200km on a bearing of 125°. Find: 1. The distance AC to nearest kin. 2. The bearing of C from A
Note that the course change was 90°. So,
#1. AC^2 = 120^2 + 200^2
#2. The bearing from A to C is θ+35° where tanθ = 200/120
To find the distance AC, we can use the Law of Cosines because we have a triangle with two known sides and the included angle.
1. Distance AC:
Using the Law of Cosines, we have:
AC² = AB² + BC² - 2 * AB * BC * cos(angle ABC)
Given values:
AB = 120 km
BC = 200 km
angle ABC = 180° - (angle A + angle B)
angle A = 35° (given)
angle B = 125° (given)
Plugging in the values, we have:
AC² = 120² + 200² - 2 * 120 * 200 * cos(180° - (35° + 125°))
AC² = 14400 + 40000 - 2 * 120 * 200 * cos(180° - 160°)
AC² = 14400 + 40000 - 2 * 120 * 200 * cos(20°)
AC² ≈ 14400 + 40000 - 48000 * cos(20°)
AC ≈ √(14400 + 40000 - 48000 * cos(20°))
Using a calculator, AC ≈ 218.34 km (rounded to the nearest kilometer).
2. Bearing of C from A:
To find the bearing from A to C, we can use the formula:
tan(bearing) = sin(angle B) / (cos(angle B) * sin(angle A) - sin(angle B) * cos(angle A))
Plugging in the values, we have:
tan(bearing) = sin(125°) / (cos(125°) * sin(35°) - sin(125°) * cos(35°))
Using a calculator, we find that tan(bearing) ≈ 1.64493.
To find the bearing, we take the inverse tangent (tan⁻¹) of 1.64493:
bearing ≈ tan⁻¹(1.64493)
Using a calculator, we find that the bearing of C from A ≈ 57.34° (rounded to the nearest degree).
Therefore, the answers are:
1. The distance AC ≈ 218 km (rounded to the nearest kilometer).
2. The bearing of C from A ≈ 57° (rounded to the nearest degree).
To find the distance AC, we can use the cosine rule. The cosine rule states that in a triangle with sides of lengths a, b, and c, and an opposite angle C, the following equation holds true:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, let's call the length of AB as a, BC as b, and AC as c.
1. Distance AC to the nearest kilometer:
Using the cosine rule, we have:
AC^2 = AB^2 + BC^2 - 2 * AB * BC * cos(∠ABC)
AC^2 = 120^2 + 200^2 - 2 * 120 * 200 * cos(125° - 35°)
Now, we can substitute the values and calculate the result:
AC^2 = 14400 + 40000 - 48000 * cos(90°)
AC^2 = 54400 - 48000 * 0
AC^2 = 54400
Taking the square root of both sides, we get:
AC = √54400 ≈ 233.19 km (nearest kilometer)
Therefore, the distance AC to the nearest kilometer is approximately 233 km.
2. Bearing of C from A:
To find the bearing of C from A, we can use trigonometry. The bearing is measured in relation to the north direction, clockwise from the north, and is usually given as a three-digit number.
In this case, the bearing is given as 035° between A and B, and the plane then changes direction to a bearing of 125° from B to C.
To find the bearing of C from A, we need to calculate the angle between the north direction and line AC. This can be done using the following formula:
bearing = 90° - arctan(AC/AB)
Taking the values we know:
bearing = 90° - arctan(AC/AB)
bearing = 90° - arctan(233/120)
Using a calculator, we can find the value of the arctan expression:
arctan(233/120) ≈ 63.26°
Substituting this value into the bearing equation:
bearing = 90° - 63.26°
bearing ≈ 26.74°
Therefore, the bearing of C from A is approximately 26.74°.