fhe man and the projection of the airship to the nearest metre. From the base of a house whose height is 20 m ., it was found that the angle of elevatio of the top of a tower was of measure \( 25^{\circ} \) Then it was found for a next time that the angl of elevation of the top of the same tower from the top of the house was of measure \( 18^{\circ} \) Find the height of the tower. b) From the top of a mountain of height 100 m ., it was found that the measure of the depression angle of the top of a rock was \( 42^{\circ} \). 37 Find the height of the rock if the distance between it and the mountain was 22 m . , knowing that they are on the same

To find the height of the tower, we can use some trigonometry! First, define \( h \) as the height of the tower. When standing at the base of the house (20 m), the line of sight to the tower top creates a right triangle with an angle of elevation of \( 25^{\circ} \). Using the tangent function: \[ \tan(25^\circ) = \frac{h - 20}{d} \rightarrow d = \frac{h - 20}{\tan(25^\circ)} \] The next triangle, standing from the top of the house (at 20 m), forms an angle of elevation of \( 18^{\circ} \): \[ \tan(18^\circ) = \frac{h}{d} \rightarrow d = \frac{h}{\tan(18^\circ)} \] Set the two expressions for \( d \) equal to each other to solve for \( h \). Next, for the rock's height, we’ll use the depression angle from the mountain top. The angle of depression \( \theta = 42^\circ \) means it forms a right triangle down to the rock. If the mountain is 100 m tall, the height of the rock \( r \) can be found using: \[ \tan(42^\circ) = \frac{100 - r}{22} \] Rearranging gives: \[ 100 - r = 22 \tan(42^\circ) \rightarrow r = 100 - 22 \tan(42^\circ) \] Plug in \( \tan(42^\circ) \) to find the rock’s height. Math is fun!