Final answer:
To calculate the curved surface area of the cone, we use the formula involving the radius and slant height, resulting in approximately 2010.72 cm². The vertical angle is found using the trigonometric tan inverse function on the height and radius, yielding approximately 138.88 degrees.
Step-by-step explanation:
The student wants to calculate the curved surface area of a cone with a height of 40 cm and a base diameter of 30 cm and its vertical angle. To find the curved surface area of a cone, we use the formula CSA = πrl, where r is the radius of the base and l is the slant height of the cone. Since the diameter is given as 30 cm, we divide by 2 to get the radius r = 15 cm. The slant height can be found using the Pythagorean theorem because the radius, height, and slant height form a right-angled triangle. So, l = √(r² + h²) = √(15² + 40²) = √(225 + 1600) = √1825 = 42.72 cm. Plugging these values into the formula, we get CSA = π * 15 cm * 42.72 cm ≈ 2010.72 cm².
To determine the vertical angle of the cone, we would need trigonometry. Since we have the opposite side (height = 40 cm) and adjacent side (radius = 15 cm) of the triangle within the cone, we can use the tan inverse function: vertical angle = 2 * tan⁻¹(opposite/adjacent) = 2 * tan⁻¹(40/15) ≈ 2 * 69.44° = 138.88°.