SOLUTION: A canvas camping tent is to be constructed in the shape of a pyramid with a square base. A pole of height h = 12 feet will form the center support, as illustrated in the figure. Fi

Algebra ->  Rational-functions -> SOLUTION: A canvas camping tent is to be constructed in the shape of a pyramid with a square base. A pole of height h = 12 feet will form the center support, as illustrated in the figure. Fi      Log On


   

Question 1113494: A canvas camping tent is to be constructed in the shape of a pyramid with a square base. A pole of height h = 12 feet will form the center support, as illustrated in the figure. Find the length x of a side of the base so that the total amount of canvas needed for the sides and bottom is 864 ft^2.
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


The base is a square of side length x; its area is x^2.

The sides of the tent are four congruent triangles with base x.
The altitude of each triangle is the slant height of the pyramid.
With the square base of side length x and a center pole of 12, the slant height is the hypotenuse of a right triangle with legs x/2 and 12.

So the area of each triangular side of the tent is one-half base times height:
%281%2F2%29%28x%29%28sqrt%28%28x%2F2%29%5E2%2B12%5E2%29%29

The total surface area of the four sides and bottom of the tent is then
x%5E2%2B2x%28sqrt%28%28x%2F2%29%5E2%2B12%5E2%29%29

This total surface area is to be 864:

x%5E2%2B2x%28sqrt%28%28x%2F2%29%5E2%2B12%5E2%29%29+=+864
2x%28sqrt%28%28x%2F2%29%5E2%2B12%5E2%29%29+=+864-x%5E2
4x%5E2%28x%5E2%2F4%2B144%29+=+746496-1728x%5E2%2Bx%5E4

(Looks ugly -- but the x^4 terms will cancel, leaving a simple quadratic equation...)

x%5E4%2B576x%5E2+=+746496-1728x%5E2%2Bx%5E4
2304x%5E2+=+746496
x%5E2+=+324
x+=+18

Answer: The side length of the tent is 18 feet.

(by the way... I don't think this is a CAMPING tent!!)