Final answer:
Tripling all dimensions of a cone results in the surface area increasing by a factor of 9, not simply tripling. This occurs because the formula for surface area, which includes the base area and lateral surface, inherently squares the radius, leading to a quadratic relationship with size changes.
Step-by-step explanation:
If you triple all dimensions of a cone, the surface area does not simply triple. To understand why, we need to look at the formula for the surface area of a cone, which is given by the sum of the base area and the lateral surface area. The formula for the surface area S of a cone is:
S = πr^2 + πrl
where r is the radius of the base and l is the slant height of the cone.
When you triple each dimension (the radius r and the slant height l), these dimensions are now 3r and 3l, respectively. Substituting these into the formula gives:
S' = π(3r)^2 + π(3r)(3l)
S' = 9πr^2 + 9πrl
Clearly, the new surface area S' is 9 times the original base area plus 9 times the original lateral surface area, not three times the original total surface area. So, if you triple all dimensions of a cone, its surface area increases by a factor of 9, not 3.
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