Q:

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

Accepted Solution

A:
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 = 802. 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=290You 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.