1. **Problem 6**: A cone has a horizontal base diameter of 24 cm and an axis length of 36 cm with a volume of 3000 cm³. Find the angle of inclination of the axis with the horizontal in degrees. Step 1: Calculate the base radius $r$. $$r = \frac{24}{2} = 12\text{ cm}$$ Step 2: Use the volume formula for a cone: $$V = \frac{1}{3} \pi r^2 h$$ Given $V = 3000$ cm³, solve for height $h$: $$3000 = \frac{1}{3} \pi (12)^2 h$$ $$3000 = \frac{1}{3} \pi \times 144 \times h$$ $$3000 = 48\pi h$$ $$h = \frac{3000}{48\pi} = \frac{3000}{150.796} \approx 19.89\text{ cm}$$ Step 3: The axis length $a = 36$ cm is the slant height of the cone. Step 4: Let $\theta$ be the angle of inclination of the axis with the horizontal. Since the height is vertical and the axis is the slant height, $$\sin \theta = \frac{h}{a} = \frac{19.89}{36} \approx 0.5525$$ Step 5: Find $\theta$: $$\theta = \arcsin(0.5525) \approx 33.5^\circ$$ **Answer:** The angle of inclination is approximately $33.5^\circ$. --- 2. **Problem 7**: The slant height of a right circular cone is 5 m, and the base diameter is 6 m. Find the volume. Step 1: Calculate the base radius: $$r = \frac{6}{2} = 3\text{ m}$$ Step 2: Let the height be $h$ and slant height $l = 5$ m. Using the Pythagorean theorem in the right triangle formed by height, radius, and slant height: $$l^2 = h^2 + r^2$$ $$5^2 = h^2 + 3^2$$ $$25 = h^2 + 9$$ $$h^2 = 16$$ $$h = 4\text{ m}$$ Step 3: Calculate the volume: $$V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (3)^2 (4) = \frac{1}{3} \pi \times 9 \times 4 = 12\pi \approx 37.70\text{ m}^3$$ **Answer:** The volume of the cone is approximately $37.70$ cubic meters. --- 3. **Problem 8**: Find the volume of a cone constructed from a sector with diameter 72 cm and central angle 210°. Step 1: The radius of the original circle is: $$R = \frac{72}{2} = 36\text{ cm}$$ Step 2: The length of the arc for the sector (which becomes the base circumference of the cone) is: $$L = 2\pi R \times \frac{210}{360} = 2\pi \times 36 \times \frac{7}{12} = 42\pi\text{ cm}$$ Step 3: The circumference of the cone base is $L = 42\pi$, so the base radius $r$ of the cone is: $$2\pi r = 42\pi$$ $$r = 21\text{ cm}$$ Step 4: The slant height $l$ of the cone equals the radius of the original circle: $$l = 36\text{ cm}$$ Step 5: Find the height $h$ of the cone using Pythagoras: $$l^2 = h^2 + r^2$$ $$36^2 = h^2 + 21^2$$ $$1296 = h^2 + 441$$ $$h^2 = 1296 - 441 = 855$$ $$h = \sqrt{855} \approx 29.24\text{ cm}$$ Step 6: Calculate the volume: $$V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (21)^2 (29.24) = \frac{1}{3} \pi \times 441 \times 29.24 \approx 1343 \pi \approx 4218.5\text{ cm}^3$$ **Answer:** The volume of the cone is approximately $4218.5$ cubic centimeters.