Triangulation can be used to find the location of an object by measuring the angles to the
object from two points at the end of a baseline. Two lookouts 20 miles apart on the coast
spot a ship at sea. Using the figure below find the distance, d, the ship is from shore to the
nearest tenth of a mile.
To find the distance, d, of the ship from the shore using triangulation, we can use the concept of similar triangles. Here's how you can solve it:
Step 1: Label the given information in the figure:
- Let A be the position of the left lookout on the coast.
- Let B be the position of the right lookout on the coast.
- Let S be the position of the ship.
- Let C be the point directly under the ship on the coast.
Step 2: Draw a line segment from A to C to represent the perpendicular from the ship to the baseline of the lookouts. Similarly, draw a line segment from B to C.
Step 3: Use the given information to mark the angles:
- The angle formed by the baseline and the line segment from A to C is labeled as α.
- The angle formed by the baseline and the line segment from B to C is labeled as β.
- The angle formed by the line segment from A to C and the line segment from S to C is labeled as γ.
- The right angle formed by the baseline and the perpendiculars from the lookouts to the baseline is labeled as 90 degrees.
Step 4: Apply the concept of similar triangles:
- Since triangle ABC is a right triangle, we can use the tangent function to relate the angles and distances.
- tan(α) = (d / AC) [1] (angle α is opposite side d and adjacent side AC)
- tan(β) = (d / BC) [2] (angle β is opposite side d and adjacent side BC)
- tan(γ) = (20 / AC) [3] (angle γ is opposite side 20 and adjacent side AC)
Step 5: Solve the equations to find the value of d:
- Divide equation 1 by equation 2:
tan(α) / tan(β) = (d / AC) / (d / BC)
tan(α) / tan(β) = BC / AC
tan(α) / tan(β) = (20 - BC) / AC [since AC + BC = 20]
tan(α) / tan(β) = (20 - d) / AC [using equation 2, d / BC = tan(β)]
- Substitute equation 3 into the previous equation:
tan(α) / tan(β) = (20 - d) / (20 / tan(γ))
tan(α) = tan(β) * (20 - d) * tan(γ) / 20
- Since α and β are given in the problem, we can find the value of d using the equation above.
Step 6: Calculate the value of d:
- Plug in the known values of the angles:
tan(33) = tan(57) * (20 - d) * tan(γ) / 20
- Solve this equation for d using algebraic manipulation.
Using this process, you can find the distance, d, of the ship from the shore to the nearest tenth of a mile.