1. Problem: A cone is 8 cm high and its vertical angle is 62°. Find the diameter of its base. 2. Formula: The vertical angle of the cone is the angle at the apex of the isosceles triangle formed by the height and radius. Half the vertical angle is 31°. Using right triangle trigonometry, $\tan(31^\circ) = \frac{r}{h}$ where $r$ is the radius and $h=8$ cm. 3. Calculation: $r = h \tan(31^\circ) = 8 \times \tan(31^\circ) \approx 8 \times 0.6009 = 4.807$ cm. 4. Diameter $d = 2r = 2 \times 4.807 = 9.614$ cm. 1. Problem: An isosceles triangle has a vertical angle of 116° and its base is 8 cm long. Calculate its height. 2. Formula: Split the triangle into two right triangles by bisecting the base. Each right triangle has angle $\frac{116^\circ}{2} = 58^\circ$ and half the base $4$ cm. 3. Height $h = 4 \tan(58^\circ) \approx 4 \times 1.6003 = 6.401$ cm. 1. Problem: Find the angle of elevation of the top of a flagpole 31.9 m high from a point 55 m away on level ground. 2. Formula: $\tan(\theta) = \frac{\text{height}}{\text{distance}} = \frac{31.9}{55}$. 3. Calculation: $\theta = \tan^{-1}(\frac{31.9}{55}) \approx \tan^{-1}(0.58) = 30.1^\circ$. 1. Problem: The gradient of a road is 1 vertically in 4 horizontally. Calculate the angle the road makes with the horizontal. 2. Formula: $\tan(\theta) = \frac{1}{4} = 0.25$. 3. Calculation: $\theta = \tan^{-1}(0.25) = 14.04^\circ$. 1. Problem: From a point on level ground 40 m away, the angle of elevation of the top of a tree is 32.5°. Calculate the height of the tree. 2. Formula: $\tan(32.5^\circ) = \frac{h}{40}$. 3. Calculation: $h = 40 \times \tan(32.5^\circ) \approx 40 \times 0.636 = 25.44$ m. 1. Problem: If the tree is 21.6 m high and the distance is 40 m, find the angle of elevation. 2. Formula: $\tan(\theta) = \frac{21.6}{40} = 0.54$. 3. Calculation: $\theta = \tan^{-1}(0.54) = 28.3^\circ$. Final answers: - Diameter of cone base: 9.61 cm - Height of isosceles triangle: 6.40 cm - Angle of elevation to flagpole: 30.1° - Road angle with horizontal: 14.04° - Height of tree: 25.44 m - Angle of elevation to tree: 28.3°