Answer Key 4.5

  1. L=2W-3 \text{ and } P=2L+2W \Rightarrow 54=2(2W-3)+2W
  2. L=2W-8 \text{ and } P=2L+2W \Rightarrow 64=2(2W-8)+2W
  3. L=2W+4 \text{ and } P=2L+2W \Rightarrow 32=2(2W+4)+2W
  4. A_1=2A_2, A_1=10^{\circ}+A_3, A_1+A_2+A_3=180^{\circ} \Rightarrow
    A_1+\dfrac{A_1}{2}+A_1- 10^{\circ}=180^{\circ}
  5. A_1=\dfrac{1}{2}A_2, A_1=20^{\circ}+A_3, A_1+A_2+A_3=180^{\circ} \Rightarrow
    A_1+2A_1+A_1-20^{\circ}=180^{\circ}
  6. A_1+A_2=\dfrac{1}{2}A_3, A_1+A_2+A_3=180^{\circ} \Rightarrow
    \dfrac{3}{2}A_3=180^{\circ}\hspace{0.34in} A_1 \text{ and } A_2?
  7. x_1+x_2=140, x_1=5x_2 \Rightarrow 5x_2+x_2=140
  8. x_1+x_2=48, x_2=5+x_1 \Rightarrow x_1+5+x_1=48
  9. \begin{array}{rrrrrrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ A_2&=&A_1&&&&&&&&&& \\ A_3&=&A_1&+&12&&&&&&&& \\ \\ A_1&+&A_2&+&A_3&&&=&180&&&& \\ A_1&+&A_1&+&A_1&+&12&=&180&&&& \\ &&&&&-&12&&-12&&&& \\ \midrule &&&&&&3A_1&=&168&&&& \\ \\ &&&&&&A_1&=&\dfrac{168}{3}&=&56&& \\ A_1&=&56^{\circ}&&&&&&&&&& \\ A_2&=&56^{\circ}&&&&&&&&&& \\ A_3&=&56^{\circ}&+&12^{\circ}&=&68^{\circ}&&&&&& \end{array}
  10. \begin{array}{rrrrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ A_1&=&A_2&&&&&&&& \\ A_3&=&A_1&-&12&&&&&& \\ \\ A_1&+&A_2&+&A_3&&&=&180&& \\ A_1&+&A_1&+&A_1&-&12&=&180&& \\ &&&&&+&12&&+12&& \\ \midrule &&&&&&3A_1&=&192&& \\ \\ &&&&&&A_1&=&\dfrac{192}{3}&=&64 \\ A_1&=&64^{\circ}&&&&&&&& \\ A_2&=&64^{\circ}&&&&&&&& \\ A_3&=&64^{\circ}&-&12^{\circ}&=&52^{\circ}&&&& \end{array}
  11. \begin{array}{rrlrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ A_1&=&A_2&&&&&&&& \\ A_3&=&3A_1&&&&&&&& \\ \\ A_1&+&A_2&+&A_3&=&180&&&& \\ A_1&+&A_1&+&3A_1&=&180&&&& \\ &&&&5A_1&=&180&&&& \\ \\ &&&&A_1&=&\dfrac{180}{5}&=&36&& \\ A_1&=&36^{\circ}&&&&&&&& \\ A_2&=&36^{\circ}&&&&&&&& \\ A_3&=&3(36^{\circ})&=&108^{\circ}&&&&&& \end{array}
  12. \begin{array}{rrrcrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ A_2&=&2A_1&&&&&&&& \\ A_3&=&A_1&+&20&&&&&& \\ \\ A_1&+&A_2&+&A_3&&&=&180&& \\ A_1&+&2A_1&+&A_1&+&20&=&180&& \\ &&&&&-&20&=&-20&& \\ \midrule &&&&&&4A_1&=&160&& \\ \\ &&&&&&A_1&=&\dfrac{160}{4}&=&40 \\ A_1&=&40^{\circ}&&&&&&&& \\ A_2&=&2(40^{\circ})&=&80^{\circ}&&&&&& \\ A_3&=&20^{\circ}&+&40^{\circ}&=&60^{\circ}&&&& \end{array}
  13. \begin{array}{rrlllrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ ^{\text{1}}L&=&W&+&15&& \\ \\ ^{\text{2}}P&=&2L&+&2W&& \\ 150&=&2(W&+&15)&+&2W \\ 150&=&2W&+&30&+&2W \\ -30&&&-&30&& \\ \midrule 120&=&4W&&&& \\ \\ W&=&\dfrac{120}{4}&=&30\text{ cm}&& \\ \\ ^{\text{3}}L&=&30&+&15&& \\ L&=&45\text{ cm}&&&& \\ \end{array}
  14. \begin{array}{rrlllrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ ^{\text{1}}L&=&W&+&40&& \\ \\ ^{\text{2}}P&=&2L&+&2W&& \\ 304&=&2(W&+&40)&+&2W \\ 304&=&2W&+&80&+&2W \\ -80&&&-&80&& \\ \midrule 224&=&4W&&&& \\ \\ W&=&\dfrac{224}{4}&=&56\text{ cm}&& \\ \\ ^{\text{3}}L&=&56&+&40&& \\ L&=&96\text{ cm}&&&& \\ \end{array}
  15. \begin{array}{rrlllrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ ^{\text{1}}W&=&L&-&22&& \\ \\ ^{\text{2}}P&=&2L&+&2W&& \\ 152&=&2L&+&2(L&-&22) \\ 152&=&2L&+&2L&-&44 \\ +44&&&&&+&44 \\ \midrule 196&=&4L&&&& \\ \\ L&=&\dfrac{196}{4}&=&49\text{ m}&& \\ \\ ^{\text{3}}W&=&49&-&22&& \\ L&=&27\text{ m}&&&& \\ \end{array}
  16. \begin{array}{rrlllrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ ^{\text{1}}W&=&L&-&26&& \\ \\ ^{\text{2}}P&=&2L&+&2W&& \\ 280&=&2L&+&2(L&-&26) \\ 280&=&2L&+&2L&-&52 \\ +52&&&&&+&52 \\ \midrule 332&=&4L&&&& \\ \\ L&=&\dfrac{332}{4}&=&83\text{ m}&& \\ \\ ^{\text{3}}W&=&83&-&26&& \\ L&=&57\text{ m}&&&& \\ \end{array}
  17. \begin{array}{rrrrrrrr} \\ \\ \\ \\ \\ x&+&2x&=&12&&& \\ &&3x&=&12&&& \\ \\ &&x&=&\dfrac{12}{3}&=&4\text{ cm}& \\ \\ &\therefore &2x&=&2(4)&=&8\text{ cm}& \\ \end{array}\text{Pieces are 4 cm and 8 cm}

  18. \begin{array}{rrrrrrrrcrr} \\ \\ \\ \\ \\ \\ \\ x&+&x&+&2&=&30&&&& \\ &&&-&2&&-2&&&& \\ \midrule &&&&2x&=&28&&&& \\ \\ &&&&x&=&\dfrac{28}{2}&=&14\text{ m}&& \\ \\ &\therefore &x&+&2&=&14&+&2&=&16 \text{ m} \end{array}\text{Pieces are 14 m and 16 m}

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