SOLUTION: How far from the top must you cut a conical tent in order to cut the cloth in half?

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Question 1193435: How far from the top must you cut a conical tent in order to cut the cloth in half?
Found 2 solutions by greenestamps, Edwin McCravy:
Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!


The amount of cloth required is proportional to the surface area of the tent.

Assuming the cut is made so that the smaller tent is similar to the original tent, the ratio of surface areas is the square of the ratio of the heights.

So, given that the ratio of the surface areas is 1/2; the ratio of heights needs to be the square root of 1/2:

= 0.707 to 3 decimal places

ANSWER: To cut the amount of cloth in half, if the height of the tent is h, then the cut must be made 0.707h from the top of the tent.
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!
If the (linear) scale factor of two similar solids is k, then
(a) all corresponding length properties of the two solids have scale factor k.
(b) all corresponding area properties of the two solids have scale factor k2.
(c) all corresponding volume properties of the two solids have scale factor k3.



Let k be the (linear) scale factor of similar cones ABC and ADE.

Then 

We are given that the corresponding surface areas have scale factor 2.

Therefore by (b) above, the scale factor of the surface areas is 

k2 = 2. Therefore,

 and 


  


Taking reciprocals of both sides,

 which is approximately 0.707,

So to cut the cloth of the tent in half, you would cut it straight
across from B to C, which is about 70.7% of its height from the top.

Edwin
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