Final answer:
The original surface area of the cone is 616.3 cm². When the diameter and the slant height are halved, the new surface area becomes 154.1 cm², which is four times smaller than the original.
Step-by-step explanation:
To find the surface area of a cone, we need to calculate the area of the base (which is a circle) and the lateral area (the area of the cone's slanted side). The formula for the surface area (SA) of a cone is given by SA = πr² + πrl, where r is the radius and l is the slant height of the cone.
The initial diameter of the cone is 14 cm, which gives us a radius of 7 cm (since the radius is half of the diameter). Using 3.14 for π, we calculate the base area as πr² = 3.14 × 7² = 153.86 cm². The slant height is given as 21 cm, so the lateral area is πrl = 3.14 × 7 × 21 = 462.42 cm². The combined surface area is therefore 153.86 cm² + 462.42 cm² = 616.28 cm², which can be rounded to 616.3 cm² to match the two significant figures of the given dimensions.
When the diameter and the slant height are both halved, the radius becomes 3.5 cm and the new slant height becomes 10.5 cm. The new base area is πr² = 3.14 × 3.5² = 38.465 cm², and the new lateral area is πrl = 3.14 × 3.5 × 10.5 = 115.605 cm², leading to a new surface area of 38.465 cm² + 115.605 cm² = 154.07 cm², which we can round to 154.1 cm² again following significant figure rules.
Thus, when the dimensions are halved, the new surface area is four times smaller because area scales with the square of the linear dimensions.