From the bottom and from the top of a 15 m tall building, the angles of elevation of the top of a nearby tower are 45° and 30° respectively. The height of the tower is:

Let the tower be H m tall and d m away. From the foot of the building H = d·tan45° = d. From its top, H − 15 = d·tan30° = d/√3. Substituting d = H gives H − H/√3 = 15, so H(√3 − 1) = 15√3 and H = 15√3/(√3 − 1) = 15(3 + √3)/2 ≈ 35.5 m. Rationalising with the wrong conjugate produces 15(3 − √3)/2 ≈ 9.5 m, which is shorter than the building itself.