Leia cuts congruent triangular patches with an area of 45 square centimeters from a rectangular piece of fabric that 18 centimeters long and 10 centimeters wide. How many of the patches can Leia cut from 32 pieces of the fabric?

1 Answer
SCooke
Apr 19, 2018

In BOTH optimum dimension cases we have a maximum of 16 triangles per sheet. 32 sheets will provide 512 triangles.

Explanation:

This has two parts. 1. What is the maximum number of congruent (equal-dimensioned) triangles with an area of #45 cm^2# that an be cut from a rectangular sheet of dimensions #18cm xx 10cm#? 2. How many pieces can be obtained from 32 pieces?

Congruent right triangles (forming rectangles in pairs) would seem to be the optimum shape, as others would leave more scrap.

A right triangle with an area of #45 cm^2# will have dimensions such that:
#45 = (1/2)xxbxxh#; or #bxxh = 22.5#
We will first look at the exact use of one side length, then the other of the stack sheet.
For a triangle height of 10cm, the base would be 2.25cm.
#18/2.25 xx 2 = 16 triangles, exactly. NO scrap!

For a triangle height of 18cm, the base would be 1.25cm.
#10/1.25 xx 2 = 16 triangles also, NO scrap!
ANY other dimension with an area of 45 will result in scrap, so one of these two dimensions must be used. In BOTH cases we have a maximum of 16 triangles per sheet. 32 sheets will provide 512 triangles.

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