Question 1116875
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The lateral surface area is {{{(pi)(r)(l)}}} where l is the slant height.<br>
With a radius of 10 and a height of 30, the slant height is {{{sqrt(10^2+30^2) = sqrt(1000) = 10sqrt(10)}}}<br>
Then the lateral surface area is {{{(pi)(10)(10sqrt(10)) = 100pi*sqrt(10)}}}<br>
Think of the cross section 20cm from the vertex as the base of a cone with the same vertex as the original cone and a height of 20cm instead of 30cm.  The two cones will be similar, with a scale factor of 20/30 = 2/3.  That means the ratio of corresponding areas will be 2^2/3^2 = 4/9.  So the area of the cross section 20cm from the vertex is {{{100pi*(4/9) = 400pi/9}}}.