Question

 

why is the slant height s = sqrt(2) - x . Please explain thoroughly how to find the slant height.

(Folding a pyramid) A pyramid with a square base and four faces, each in the shape of an isosceles triangle, is made by cutting away four triangles from 2 ft square piece of cardboard and bending up the resulting triangles to form the walls of the pyramid. What is the largest volume the pyramid can have? Hint: The volume of a pyramid having base area A and height h measured perpendicular to the base is V = (1/3)Ah.

Answer 1

Approach: While dealing with these kind of problems when you are facing difficulty to understand the relations between the sides, just try to observe and make the relations between the given quantities through all possible methods. In case of triangles or more generally when you are dealing with polygons you can think of pythagoras thorem, Similar and Congruent Triangles, Trigonometric functions, or sometimes like in this case simple length balancing techniques may work.

Now, consider the following figure, related to your problem.

In the above figure,

s is the required slant height of pyramid,

2x is the side of the square base,

L is the side length of square cardboard,

& ABCD is the square base.

Just observe the length of the diagonal of the complete cardboard and write it using two lengths. Thus, we can write that,

Side length of square base + 2 times slant height of pyramid = Length of Diagonal of the original cardboard

Thus, 2s+ 2x= 2^1/2 .L

Also, L=2 ft given in the problem

2s + 2x= 2.2^1/2

s=2^1/2 - x