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A box of wood pieces contains wood cut into triangular, square, and
pentagonal shapes. There are 80 pieces of wood in the box, and the pieces
have a total of 290 sides. If there are 10 more triangular pieces than square
pieces, find the number of pieces of wood of each shape in the box.

Respuesta :

frika

Answer:

There are 30 square pieces, 40 triangular pieces and 10 pentagonal pieces.

Step-by-step explanation:

Let x be the number of square pieces. If there are 10 more triangular pieces than square pieces, then there are (x + 10) triangular pieces. Let y be the number of pentagonal pieces.

1. There are 80 pieces of wood in the box, then

(x + 10) + x + y = 80

2. There are

  • 3 sides in a triangular piece [tex]\rightarrow[/tex] there are 3(x + 10) sides in x + 10 pieces;
  • 4 sides in a square piece [tex]\rightarrow[/tex] there are 4x sides in x pieces;
  • 5 sides in a pentagonal piece [tex]\rightarrow[/tex] there are 5y sides in y pieces;
  • in total, there are 290 sides.

So,

3(x+10)+4x+5y=290

You get the system of two equations:

[tex]\left\{\begin{array}{l}(x+10)+x+y=80\\3(x+10)+4x+5y=290\end{array}\right.\Rightarrow \left\{\begin{array}{l}2x+y=70\\7x+5y=260\end{array}\right.[/tex]

From the first equation,

[tex]y=70-2x[/tex]

Substitute it into the second equation

[tex]7x+5(70-2x)=260\\ \\7x+350-10x=260\\ \\-3x=260-350\\ \\-3x=-90\\ \\x=30\\ \\x+10=40\\ \\y=70-2\cdot 30=10[/tex]

Hence, there are 30 square pieces, 40 triangular pieces and 10 pentagonal pieces.

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