Final answer:
The conical-shaped lid has a volume of approximately 29.32 cm³ and a surface area of about 60.35 cm². The dimensions of the smallest possible pyramid-shaped box to package the bottle with its lid are a base of 5 cm by 5 cm and a height of 7 cm.
Step-by-step explanation:
To calculate the volume and surface area of the conical-shaped lid, we can use the following formulas:
- Volume of a cone (V) = (1/3)πr²h
- Surface area of a cone (A) = πr(r + √(r² + h²))
First, we convert the diameter of the lid to radius by dividing it by 2, giving us r = 2.5 cm. With a height (h) of 4.5 cm, we have:
- V = (1/3)π(2.5 cm)²(4.5 cm) = π(6.25 cm²)(4.5 cm) / 3 ≈ 29.32 cm³
- A = π(2.5 cm)(2.5 cm + √((2.5 cm)² + (4.5 cm)²)) ≈ π(2.5 cm)(2.5 cm + 5.1 cm) ≈ 60.35 cm²
To determine the dimensions of the pyramid-shaped box, we must ensure that the pyramid is large enough to encompass the total height of the spherical bottle and the conical lid. Assuming the diameter of the sphere is at least 5 cm (the same as the lid), the total height of the bottle and lid together would be the radius of the sphere plus the height of the cone.
Since the diameter of the sphere is equal to that of the lid, the radius would be 2.5 cm, giving us a total height of 2.5 cm + 4.5 cm = 7 cm for the pyramid. The minimum base of the pyramid would then be a square with sides equal to the diameter of the sphere, which is 5 cm.Thus, the dimensions of the smallest possible pyramid-shaped box would have a base of 5 cm × 5 cm and a height of 7 cm.