A lighthouse sits at the edge of a cliff, as shown. A ship at sea level is 825 meters from the base of the cliff. The angle of elevation from sea level to the base of the lighthouse is $37.2^{\circ}$. The angle of elevation from sea level to the top of the lighthouse is $38.3^{\circ}$. Find the height of the lighthouse from the top of the cliff.
Do not round any intermediate computations. Round your answer to the nearest tenth.
Note that the figure below is not drawn to scale.
Answer
Rounding to the nearest tenth, we find that the height of the lighthouse from the top of the cliff is approximately \(\boxed{25.3}\) meters.
Step 1 ：Given that the distance from the ship to the base of the cliff is 825 meters, the angle of elevation from sea level to the base of the lighthouse is \(37.2^{\circ}\), and the angle of elevation from sea level to the top of the lighthouse is \(38.3^{\circ}\).
Step 2 ：We can use the tangent of the angles of elevation to find the height of the lighthouse. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
Step 3 ：Setting up two equations using the tangent of the two angles of elevation, the first equation will give us the height from sea level to the base of the lighthouse and the second equation will give us the height from sea level to the top of the lighthouse.
Step 4 ：Using the tangent of the angle of elevation to the base of the lighthouse, we find that the height from sea level to the base of the lighthouse is approximately 626.2090832679901 meters.
Step 5 ：Using the tangent of the angle of elevation to the top of the lighthouse, we find that the height from sea level to the top of the lighthouse is approximately 651.5457402006926 meters.
Step 6 ：The difference between these two heights will be the height of the lighthouse, which is approximately 25.336656932702567 meters.
Step 7 ：Rounding to the nearest tenth, we find that the height of the lighthouse from the top of the cliff is approximately \(\boxed{25.3}\) meters.
