Four ships A,B,C and D are at sea in the following relative positions: B is on the straight line segment AC, B is due north of D and D is due west of C. The distance B and D is 2km. `angleBDA = 40^(@), angleBCD = 25^(@)`. What is the distance between A and D. (`sin 25^(@) = 0.423`).

Answer

To solve the problem, we need to find the distance between ships A and D based on the given information about their relative positions and angles. Let's break this down step by step.

### Step 1: Understanding the Positions and Angles
- We know that B is on the straight line segment AC.
- B is due north of D, and D is due west of C.
- The distance between B and D (BD) is given as 2 km.
- The angles are given as ∠BDA = 40° and ∠BCD = 25°.

### Step 2: Determine Angle ADC
Since BDA is 40° and DAB is a right angle (90°), we can find angle ADC:
- ∠ADC = ∠BDA + 90° = 40° + 90° = 130°.

### Step 3: Determine Angle DCA
We know that ∠BCD = 25°. Therefore, angle DCA can be calculated as:
- ∠DCA = 25°.

### Step 4: Calculate Angle DAC
Using the angle sum property of triangles, we can find angle DAC in triangle ADC:
- ∠DAC + ∠ADC + ∠DCA = 180°.
- ∠DAC + 130° + 25° = 180°.
- ∠DAC = 180° - 155° = 25°.

### Step 5: Apply the Property of Triangles
Since ∠DAC = ∠DCA, we can conclude that:
- AD = CD (by the property of triangles where sides opposite equal angles are equal).

### Step 6: Use Triangle BDC to Find AD
In triangle BDC, we can use the cotangent function:
- cot(25°) = CD / BD.
- Since BD = 2 km, we have:
- CD = BD * cot(25°).

### Step 7: Express cot(25°) in Terms of sin(25°)
We know that:
- cot(θ) = cos(θ) / sin(θ).
Thus:
- CD = 2 * (cos(25°) / sin(25°)).

### Step 8: Calculate cos(25°)
Using the Pythagorean identity:
- cos(25°) = √(1 - sin²(25°)).
- Given sin(25°) = 0.423, we calculate:
- sin²(25°) = (0.423)² = 0.179.
- cos(25°) = √(1 - 0.179) = √(0.821) ≈ 0.905.

### Step 9: Substitute Values to Find AD
Now we can substitute the values into the equation for CD:
- CD = 2 * (0.905 / 0.423).
- CD ≈ 2 * 2.139 = 4.278 km.

Since AD = CD, we have:
- AD ≈ 4.278 km.

### Final Answer
The distance between A and D is approximately **4.278 km**.

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