Question 1172878
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The lateral surface area of a cone is {{{(pi)(r)(l)}}}, where r is the radius of the base and <i>l</i> is the slant height.<br>
The area of the base is {{{(pi)(r^2)}}}.<br>
The intent of the question is not clear....<br>
(1) If only the slant height of the cone is doubled (the radius of the cone is unchanged), then the LATERAL surface area is doubled.  The (total) surface area of the cone -- which is literally what the question asks for, is NOT doubled, because the surface area of the base does not change.<br>
(2) If the "new" cone is similar to the original -- which means the slant height and radius are both doubled, then we have two similar solids with scale factor 2; the ratio of surface areas is 4:1.  So in this case the surface area is multiplied by a factor of 4, not 2.<br>
ANSWER: Yes, the surface area is doubled -- IF only the slant height is doubled AND by "surface area" you are only talking about LATERAL surface area.<br>