{"id":2000,"date":"2021-12-02T19:40:24","date_gmt":"2021-12-03T00:40:24","guid":{"rendered":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/answer-key-10-7\/"},"modified":"2022-11-02T10:39:00","modified_gmt":"2022-11-02T14:39:00","slug":"answer-key-10-7","status":"publish","type":"back-matter","link":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/answer-key-10-7\/","title":{"raw":"Answer Key 10.7","rendered":"Answer Key 10.7"},"content":{"raw":"
    \n \t
  1. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrrrl}\nx&+&y&=&22&\\Rightarrow &x&=&22&-&y \\\\\nx&-&y&=&120&&&&&& \\\\ \\\\\n(22&-&y)y&=&120&&&&&& \\\\\n22y&-&y^2&=&120&&&&&& \\\\ \\\\\n&&0&=&y^2&-&22y&+&120&& \\\\\n&&0&=&y^2&-&12y&-&10y&+&120 \\\\\n\\hline\n&&0&=&y(y&-&12)&-&10(y&-&12) \\\\\n&&0&=&(y&-&12)(y&-&10)&& \\\\ \\\\\n&&y&=&12,&10&&&&&\n\\end{array}[\/latex]\n[latex]\\therefore \\text{ numbers are }10, 12[\/latex]<\/li>\n \t
  2. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrccrrrrrrrrl}\n&&&&x&-&y&=&4&\\Rightarrow &x&=&y&+&4 \\\\\n&&&&x&\\cdot &y&=&140&&&&&& \\\\ \\\\\n&&&&(y&+&4)y&=&140&&&&&& \\\\\n&&&&y^2&+&4y&=&140&&&&&& \\\\ \\\\\n&&y^2&+&4y&-&140&=&0&&&&&& \\\\\ny^2&-&10y&+&14y&-&140&=&0&&&&&& \\\\\n\\hline\ny(y&-&10)&+&14(y&-&10)&=&0&&&&&& \\\\\n&&(y&-&10)(y&+&14)&=&0&&&&&& \\\\ \\\\\n&&&&&&y&=&10,&-14&&&&& \\\\ \\\\\n&&&&&&y&=&10,&x&=&10&+&4&= 14 \\\\\n&&&&&&y&=&-14,&x&=&-14&+&4&= -10 \\\\\n\\end{array}[\/latex]\n[latex]\\therefore \\text{ numbers are }10, 14\\text{ and }-10, -14[\/latex]<\/li>\n \t
  3. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrcrrrrrrrrrl}\n&&&&A&-&B&=&8&\\Rightarrow &A&=&B&+&8 \\\\\n&&&&A^2&+&B^2&=&320&&&&&& \\\\ \\\\\n&&(B&+&8)^2&+&B^2&=&320&&&&&& \\\\\nB^2&+&16B&+&64&+&B^2&=&320&&&&&& \\\\\n&&&-&320&&&&-320&&&&&& \\\\\n\\hline\n&&2B^2&+&16B&-&256&=&0&&&&&& \\\\\n&&2(B^2&+&8B&-&128)&=&0&&&&&& \\\\\n&&2(B&+&16)(B&-&8)&=&0&&&&&& \\\\ \\\\\n&&&&&&B&=&-16,&8&&&&& \\\\ \\\\\n&&&&&&\\therefore A&=&B&+&8&&&& \\\\\n&&&&&&A&=&-8,&16&&&&&\n\\end{array}[\/latex]\n[latex]\\therefore (-16, -8)\\text{ and }(8,16)[\/latex]<\/li>\n \t
  4. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrcrrrrrrrrr}\nx, &x&+&2&&&&&&&&&& \\\\ \\\\\n&&x^2&+&(x&+&2)^2&=&244&&&&& \\\\\nx^2&+&x^2&+&4x&+&4&=&244&&&&& \\\\\n&&&&&&-244&&-244&&&&& \\\\\n\\hline\n&&2x^2&+&4x&-&240&=&0&&&&& \\\\\n&&2(x^2&+&2x&-&120)&=&0&&&&& \\\\\n&&2(x&-&10)(x&+&12)&=&0&&&&& \\\\ \\\\\n&&&&&&x&=&10, &-12&&&& \\\\ \\\\\n&&&&&&x&=&10, &x&+&2&=&12 \\\\\n&&&&&&x&=&-12, &x&+&2&=&-10\\\\\n\\end{array}[\/latex]\n[latex]\\therefore \\text{ numbers are }10, 12\\text{ or }-12, -10[\/latex]<\/li>\n \t
  5. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrrrrr}\nx,&x&+&2&&&&&&&& \\\\ \\\\\n&&x^2&-&(x&+&2)^2&=&60&&& \\\\\nx^2&-&(x^2&+&4x&+&4)&=&60&&& \\\\\nx^2&-&x^2&-&4x&-&4&=&60&&& \\\\\n&&&&&+&4&&+4&&& \\\\\n\\hline\n&&&&&&\\dfrac{-4x}{-4}&=&\\dfrac{64}{-4}&&& \\\\ \\\\\n&&&&&&x&=&-16&&& \\\\ \\\\\n&&&&x&+&2&\\Rightarrow &-16&+&2& \\\\\n&&&&&&&\\Rightarrow &-14&&& \\\\\n\\end{array}[\/latex]\n[latex]-16, -14[\/latex]<\/li>\n \t
  6. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrcrrrl}\nx,&x&+&2&&&&& \\\\ \\\\\n&&x^2&+&(x&+&2)^2&=&\\phantom{-}452 \\\\\nx^2&+&x^2&+&4x&+&4&=&\\phantom{-}452 \\\\\n&&&&&-&452&&-452 \\\\\n\\hline\n&&2x^2&+&4x&-&448&=&0 \\\\\n&&2(x^2&+&2x&-&224)&=&0 \\\\\n&&2(x&-&14)(x&+&16)&=&0 \\\\ \\\\\n&&&&&&x&=&14, -16 \\\\ \\\\\n&&&&&&x&=&14 \\\\\n&&&&x&+&2&=&16 \\\\ \\\\\n&&&&&&x&=&-16 \\\\\n&&&&x&+&2&=&-14\n\\end{array}[\/latex]\n[latex]14,16\\text{ and }-16,-14[\/latex]<\/li>\n \t
  7. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrcrrrrrrrr}\nx,&x&+&2,&x&+&4&&&& \\\\ \\\\\n&&x(x&+&2)&=&38&+&x&+&4 \\\\\nx^2&+&2x&&&=&42&+&x&& \\\\\n&-&x&-&42&&-42&-&x&& \\\\\n\\hline\nx^2&+&x&-&42&=&0&&&& \\\\\n(x&+&7)(x&-&6)&=&0&&&& \\\\\n&&&&x&=&\\cancel{-7},&6&&& \\\\\n\\end{array}[\/latex]\n[latex]\\therefore \\text{ numbers are }6,8,10[\/latex]<\/li>\n \t
  8. [latex]x, x+2, x+4[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrrrr}\n&&(x)(x&+&2)&=&52&+&x&+&4 \\\\ \\\\\nx^2&+&2x&&&=&56&+&x&& \\\\\n&-&x&-&56&&-56&-&x&& \\\\\n\\hline\nx^2&+&x&-&56&=&0&&&& \\\\\n(x&+&8)(x&-&7)&=&0&&&& \\\\ \\\\\n&&&&x&=&\\cancel{-8}, 7&&&&\n\\end{array}[\/latex]\n[latex]\\therefore \\text{ numbers are }7,9,11[\/latex]<\/li>\n \t
  9. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrlrrr}\n&&A&=&T&+&4&&&&& \\\\ \\\\\nA&\\cdot &T&=&80&+&(A&-&4)&(T&-&4) \\\\\n(T&+&4)T&=&80&+&(T&+&\\cancel{4-4})&(T&-&4) \\\\ \\\\\nT^2&+&4T&=&80&+&T^2&-&4T&&& \\\\\n-T^2&+&4T&&&-&T^2&+&4T&&& \\\\\n\\hline\n&&\\dfrac{8T}{8}&=&\\dfrac{80}{8}&&&&&&& \\\\ \\\\\n&&T&=&10&&&&&&& \\\\ \\\\\n&&\\therefore A&=&T&+&4&&&&& \\\\\n&&A&=&10&+&4&=&14&&&\n\\end{array}[\/latex]<\/li>\n \t
  10. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrcrrrcrcrrrr}\n&&C&=&K&+&3&&&&&& \\\\\n&&CK&=&(C&+&5)&&(K&+&5)&-&130 \\\\ \\\\\n(K&+&3)K&=&(K&+&3&+&5)(K&+&5)&-&130 \\\\\nK^2&+&3K&=&K^2&+&13K&+&40&-&130&& \\\\\n-K^2&-&13K&&-K^2&-&13K&&&&&& \\\\\n\\hline\n&&\\dfrac{-10K}{-10}&=&\\dfrac{-90}{-10}&&&&&&&& \\\\ \\\\\n&&K&=&9&&&&&&&& \\\\ \\\\\n&&\\therefore C&=&9&+&3&=&12&&&&\n\\end{array}[\/latex]<\/li>\n \t
  11. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrcrrrrrrrr}\n&&&&&&J&=&S&+&1&& \\\\ \\\\\n&&(J&+&5)(S&+&5)&=&230&+&J&\\cdot &S \\\\\n(S&+&1&+&5)(S&+&5)&=&230&+&(S&+&1)S \\\\\n&&(S&+&6)(S&+&5)&=&230&+&S^2&+&S \\\\ \\\\\n&&S^2&+&11S&+&30&=&S^2&+&S&+&230 \\\\\n&&-S^2&-&S&-&30&&-S^2&-&S&-&30 \\\\\n\\hline\n&&&&&&\\dfrac{10S}{10}&=&\\dfrac{200}{10}&&&& \\\\ \\\\\n&&&&&&S&=&20&&&& \\\\\n&&&&&&J&=&21&&&&\n\\end{array}[\/latex]<\/li>\n \t
  12. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrcrcrrrrrrrr}\n&&&&&&J&=&S&+&2&& \\\\\n&&(S&+&2)(J&+&2)&=&48&+&S&\\cdot &J \\\\ \\\\\n(S&+&2)(S&+&2&+&2)&=&48&+&S(S&+&2) \\\\\n&&(S&+&2)(S&+&4)&=&48&+&S^2&+&25 \\\\ \\\\\n&&S^2&+&6S&+&8&=&48&+&S^2&+&25 \\\\\n&&-S^2&-&2S&-&8&&-8&-&S^2&-&25 \\\\\n\\hline\n&&&&&&\\dfrac{4S}{4}&=&\\dfrac{40}{4}&&&& \\\\ \\\\\n&&&&&&S&=&10&&&& \\\\\n&&&&&&J&=&12&&&&\n\\end{array}[\/latex]<\/li>\n \t
  13. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{ll}\n\\begin{array}[t]{rrrrrrrrl}\n&&&&&&d&=&r\\cdot t \\\\ \\\\\n&&&&r&\\cdot &t&=&240\\ \\\\\n&&&&&&\\therefore r&=&\\dfrac{240}{t} \\\\ \\\\\n&&(r&+&20)(t&-&1)&=&240 \\\\\n&&(\\dfrac{240}{t}&+&20)(t&-&1)&=&240 \\\\ \\\\\n\\cancel{240}&+&20t&-&\\dfrac{240}{t}&-&20&=&\\cancel{240} \\\\ \\\\\n&&(20t&-&\\dfrac{240}{t}&-&20&=&0)(t) \\\\ \\\\\n&&(20t^2&-&240&-&20t&=&0)(\\div 20)\n\\end{array}\n&\\hspace{0.25in}\n\\begin{array}[t]{rrcrcrl}\nt^2&-&12&-&t&=&0 \\\\\n(t&-&4)(t&+&3)&=&0 \\\\ \\\\\n&&&&t&=&4, \\cancel{-3} \\\\ \\\\\n&&&&r&=&\\dfrac{240}{4}\\text{ or }60\\text{ km\/h} \\\\ \\\\\n&&&&\\text{faster}&=&80\\text{ km\/h}\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
  14. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{ll}\n\\begin{array}[t]{rrrrrrrrl}\n&&&&&&d&=&r\\cdot t \\\\\n&&&&r&\\cdot &t&=&100 \\\\\n&&&&&&r&=&\\dfrac{100}{t} \\\\ \\\\\n&&(r&+&20)(t&-&0.5)&=&120 \\\\\n&&(\\dfrac{100}{t}&+&20)(t&-&0.5)&=&120 \\\\\n100&+&20t&-&\\dfrac{50}{t}&-&10&=&120 \\\\\n&&&&&-&120&&-120 \\\\\n\\hline\n&&20t&-&30&-&\\dfrac{50}{t}&=&0 \\\\ \\\\\n&&(20t&-&30&-&\\dfrac{50}{t}&=&0)(t) \\\\ \\\\\n&&(20t^2&-&30t&-&50&=&0)(\\div 10)\n\\end{array}\n&\\hspace{0.25in}\n\\begin{array}[t]{rrrrrrl}\n2t^2&-&3t&-&5&=&0 \\\\\n&&&&t&=&\\dfrac{-(-3)\\pm \\sqrt{(-3)^2-4(2)(-5)}}{2(2)} \\\\ \\\\\n&&&&t&=&\\dfrac{3\\pm 7}{4}=\\dfrac{10}{4}\\text{ or }\\cancel{\\dfrac{-4}{4}} \\\\ \\\\\n&&&&t&=&2.5\\text{ h}\n\\end{array}\n\\end{array}[\/latex]\n[latex]\\text{Answer: }\\dfrac{100\\text{ km}}{2.5\\text{ h}}=\\dfrac{40\\text{ km}}{\\text{h}}, \\dfrac{120\\text{ km}}{2\\text{ h}}=\\dfrac{60\\text{ km}}{\\text{h}}[\/latex]<\/li>\n \t
  15. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{ll}\n\\begin{array}[t]{rrrrrrrrl}\n&&&&&&d&=&r\\cdot t \\\\\n&&&&r&\\cdot &t&=&150 \\\\\n&&&&&&r&=&\\dfrac{150}{t} \\\\ \\\\\n&&(r&+&5)(t&-&1.5)&=&150 \\\\\n&&(\\dfrac{150}{t}&+&5)(t&-&1.5)&=&150 \\\\ \\\\\n\\cancel{150}&+&5t&-&\\dfrac{225}{t}&-&7.5&=&\\cancel{150} \\\\ \\\\\n&&(5t&-&\\dfrac{225}{t}&-&7.5&=&0)(t) \\\\ \\\\\n&&(5t^2&-&7.5t&-&225&=&0)(2) \\\\\n&&(10t^2&-&15t&-&450&=&0)(\\div 5)\n\\end{array}\n&\\hspace{0.25in}\n\\begin{array}[t]{rrrrrrl}\n2t^2&-&3t&-&90&=&0 \\\\\n(t&+&6)(2t&-&15)&=&0 \\\\\n&&&&t&=&\\cancel{-6}, \\dfrac{15}{2} \\\\ \\\\\n&&&&r&=&\\dfrac{150}{t} \\\\ \\\\\n&&&&r&=&\\dfrac{150}{\\dfrac{15}{2}} \\\\ \\\\\n&&&&r&=&\\dfrac{150}{1}\\cdot \\dfrac{2}{15} \\\\ \\\\\n&&&&r&=&20\\text{ km\/h}\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
  16. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrl}\n&&&&&&d&=&r\\cdot t \\\\\n&&&&r&\\cdot &t&=&180\\Rightarrow r=\\dfrac{180}{t} \\\\ \\\\\n&&(r&+&15)(t&-&1)&=&180 \\\\\n&&(\\dfrac{180}{t}&+&15)(t&-&1)&=&180 \\\\ \\\\\n\\cancel{180}&+&15t&-&\\dfrac{180}{t}&-&15&=&\\cancel{180} \\\\\n&&(15t&-&15&-&\\dfrac{180}{t}&=&0)(t) \\\\\n&&(15t^2&-&15t&-&180&=&0)(\\div 15) \\\\ \\\\\n&&t^2&-&t&-&12&=&0 \\\\\n&&(t&-&4)(t&+&3)&=&0 \\\\\n&&&&&&t&=&4, \\cancel{-3} \\\\ \\\\\n&&&&&&r&=&\\dfrac{180}{4}=45\n\\end{array}[\/latex]<\/li>\n \t
  17. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrcrrrl}\n&&&&r&\\cdot &t&=&72\\Rightarrow r=\\dfrac{72}{t} \\\\ \\\\\n&&(r&+&12)(9&-&t)&=&72 \\\\ \\\\\n&&(\\dfrac{72}{t}&+&12)(9&-&t)&=&72 \\\\\n\\dfrac{648}{t}&+&108&-&72&-&12t&=&72 \\\\\n&&&-&72&&&&-72 \\\\\n\\hline\n&&(-12t&-&36&+&\\dfrac{648}{t}&=&0)(t) \\\\ \\\\\n&&(-12t^2&-&36t&+&648&=&0)(\\div -12) \\\\ \\\\\n&&t^2&+&3t&-&54&=&0 \\\\\n&&(t&+&9)(t&-&6)&=&0 \\\\\n&&&&&&t&=&\\cancel{-9}, 6 \\\\ \\\\\n&&&&&&r&=&\\dfrac{72}{6}=12\\text{ (there)} \\\\ \\\\\n&&&&&&r&=&24\\text{ (return)}\n\\end{array}[\/latex]<\/li>\n \t
  18. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrcrrrl}\n&&&&r&\\cdot &t&=&120\\Rightarrow r=\\dfrac{120}{t} \\\\\n&&(r&+&10)(7&-&t)&=&120 \\\\ \\\\\n&&(\\dfrac{120}{t}&+&10)(7&-&t)&=&120 \\\\\n\\dfrac{840}{t}&+&70&-&120&-&10t&=&120 \\\\\n&&&-&120&&&&-120 \\\\\n\\hline\n&&(-10t&-&170&+&\\dfrac{840}{t}&=&0)(t) \\\\\n&&(-10t^2&-&170t&+&840&=&0)(\\div -10) \\\\ \\\\\n&&t^2&+&17t&-&84&=&0 \\\\\n&&(t&+&21)(t&-&4)&=&0 \\\\\n&&&&&&t&=&\\cancel{-21}, 4 \\\\ \\\\\n&&&&&&r&=&\\dfrac{120}{4}\\text{ or }30\\text{ km\/h} \\\\ \\\\\n&&&&r&+&10&=&40\\text{ km\/h} \\\\\n\\end{array}[\/latex]<\/li>\n \t
  19. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrcrrrl}\n&&&&r&\\cdot &t&=&240\\Rightarrow r=\\dfrac{240}{t} \\\\ \\\\\n&&(r&+&20)(t&-&1)&=&240 \\\\\n&&(\\dfrac{240}{t}&+&20)(t&-&1)&=&240 \\\\\n\\cancel{240}&+&20t&-&\\dfrac{240}{t}&-&20&=&\\cancel{240} \\\\ \\\\\n&&(20t&-&20&-&\\dfrac{240}{t}&=&0)(t) \\\\\n&&(20t^2&-&20t&-&240&=&0)(\\div 20) \\\\ \\\\\n&&t^2&-&t&-&12&=&0 \\\\\n&&(t&-&4)(t&+&3)&=&0 \\\\\n&&&&&&t&=&4, \\cancel{-3} \\\\ \\\\\n&&&&&&r&=&\\dfrac{240}{4}\\text{ or }60\\text{ km\/h}\n\\end{array}[\/latex]<\/li>\n \t
  20. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrcrrrl}\n&&&&r&\\cdot &t&=&600\\Rightarrow r=\\dfrac{600}{t} \\\\ \\\\\n&&(r&-&50)(7&-&t)&=&600 \\\\\n&&(\\dfrac{600}{t}&-&50)(7&-&t)&=&600 \\\\\n\\dfrac{4200}{t}&-&350&-&600&+&50t&=&600 \\\\\n&&&-&600&&&&-600 \\\\\n\\hline\n&&(50t&-&1550&+&\\dfrac{4200}{t}&=&0)(t) \\\\\n&&(50t^2&-&1550t&+&4200&=&0)(\\div 50) \\\\ \\\\\n&&t^2&-&31t&+&84&=&0 \\\\\n&&(t&-&3)(t&-&28)&=&0 \\\\\n&&&&&&t&=&3, \\cancel{28} \\\\ \\\\\n&&&&&&r&=&\\dfrac{600}{3}\\text{ or }200\\text{ km\/h}\n\\end{array}[\/latex]<\/li>\n \t
  21. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrlrrrl}\nL&=&4&+&W&&&& \\\\\n\\text{Area}&=&L&\\cdot &W&&&& \\\\ \\\\\n60&=&(4&+&W)W&&&& \\\\\n60&=&4W&+&W^2&&&& \\\\ \\\\\n0&=&W^2&+&4W&-&60&& \\\\\n0&=&W^2&+&10W&-&6W&-&60 \\\\\n\\hline\n0&=&W(W&+&10)&-&6(W&+&10) \\\\\n0&=&(W&+&10)(W&-&6)&& \\\\ \\\\\nW&=&\\cancel{-10},&6&&&&& \\\\\nL&=&6&+&4&=&10&&\n\\end{array}[\/latex]<\/li>\n \t
  22. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrcrrrr}\nW&=&L&-&10&&&& \\\\\n\\text{Area}&=&L&\\cdot &W&&&& \\\\ \\\\\n200&=&L(L&-&10)&&&& \\\\\n200&=&L^2&-&10L&&&& \\\\ \\\\\n0&=&L^2&-&10L&-&200&& \\\\\n0&=&L^2&+&10L&-&20L&-&200 \\\\\n\\hline\n0&=&L(L&+&10)&-&20(L&+&10) \\\\\n0&=&(L&+&10)(L&-&20)&& \\\\ \\\\\nL&=&\\cancel{-10},&20&&&&& \\\\\nW&=&20&-&10&=&10&&\n\\end{array}[\/latex]<\/li>\n \t
  23. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrcrcrcrl}\n&&&&&&\\text{Area}_{\\text{large}}&-&\\text{Area}_{\\text{small}}&=&2800\\text{ m}^2 \\\\ \\\\\n&&(150&+&2x)(120&+&2x)&-&(150)(120)&=&2800 \\\\\n\\cancel{18000}&+&240x&+&300x&+&4x^2&-&\\cancel{18000}&=&2800 \\\\\n&&&&&&&-&2800&&-2800 \\\\\n\\hline\n&&&&4x^2&+&540x&-&2800&=&0 \\\\\n&&&&x^2&+&135x&-&700&=&0 \\\\\n&&&&(x&-&5)(x&+&140)&=&0 \\\\\n&&&&&&&&x&=&5, \\cancel{-140}\n\\end{array}[\/latex]\n[latex]\\text{walkway}=5\\text{ m}[\/latex]<\/li>\n \t
  24. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrcrcrcrl}\n&&&&&&\\text{Area}_{\\text{large}}&-&\\text{Area}_{\\text{small}}&=&74\\text{ m}^2 \\\\ \\\\\n&&(25&+&2x)(10&+&2x)&-&(25)(10)&=&74 \\\\\n\\cancel{250}&+&20x&+&50x&+&4x^2&-&\\cancel{250}&=&74 \\\\\n&&&&&&&-&74&&-74 \\\\\n\\hline\n&&&&4x^2&+&70x&-&74&=&0 \\\\\n&&&&2x^2&+&35x&-&37&=&0 \\\\\n&&&&(x&-&1)(2x&+&37)&=&0 \\\\\n&&&&&&&&x&=&1, \\cancel{-\\dfrac{37}{2}} \\\\\n\\end{array}[\/latex]\n[latex]\\text{the overlap}=1\\text{ m}[\/latex]<\/li>\n \t
  25. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrl}\n&&&&L&=&W&+&4 \\\\\n&&L&\\cdot &W&=&60&& \\\\ \\\\\n&&(W&+&4)W&=&60&& \\\\\nW^2&+&4W&&&=&60&& \\\\\n&&&-&60&&-60&& \\\\\n\\hline\nW^2&+&4W&-&60&=&0&& \\\\\n(W&-&6)(W&+&10)&=&0&& \\\\ \\\\\n&&&&W&=&6,&\\cancel{-10}& \\\\\n&&&&L&=&6&+&4=10\n\\end{array}[\/latex]<\/li>\n \t
  26. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrcrr}\n&&(x&+&5)^2&=&4(x)^2&&&& \\\\ \\\\\nx^2&+&10x&+&25&=&4x^2&&&& \\\\\n-x^2&-&10x&-&25&&-x^2&-&10x&-&25 \\\\\n\\hline\n&&&&0&=&3x^2&-&10x&-&25 \\\\\n&&&&0&=&(x&-&5)(3x&+&5) \\\\\n&&&&x&=&5, &\\cancel{-\\dfrac{5}{3}}&&&\n\\end{array}[\/latex]<\/li>\n \t
  27. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrlll}\n&&&&L&=&20&+&W \\\\\n&&L&\\cdot &W&=&2400&& \\\\ \\\\\n&&(20&+&W)W&=&2400&& \\\\\nW^2&+&20W&&&=&2400&& \\\\\n&&&-&2400&&-2400&& \\\\\n\\hline\nW^2&+&20W&-&2400&=&0&& \\\\\n(W&+&60)(W&-&40)&=&0&& \\\\ \\\\\n&&&&W&=&\\cancel{-60},&40& \\\\\n&&&&L&=&20&+&40=60\n\\end{array}[\/latex]<\/li>\n \t
  28. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrcrrrrrrrr}\n&&&&&&L&=&W&+&8&& \\\\\n&&(L&+&2)(W&+&2)&=&L&\\cdot &W&+&60 \\\\ \\\\\n(W&+&8&+&2)(W&+&2)&=&(W&+&8)W&+&60 \\\\\n&&W^2&+&12W&+&20&=&W^2&+&8W&+&60 \\\\\n&&-W^2&-&8W&-&20&&-W^2&-&8W&-&20 \\\\\n\\hline\n&&&&&&\\dfrac{4W}{4}&=&\\dfrac{40}{4}&&&& \\\\ \\\\\n&&&&&&W&=&10&&&& \\\\\n&&&&&&L&=&10&+&8&=&18\n\\end{array}[\/latex]<\/li>\n<\/ol>","rendered":"
      \n
    1. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrrrl} x&+&y&=&22&\\Rightarrow &x&=&22&-&y \\\\ x&-&y&=&120&&&&&& \\\\ \\\\ (22&-&y)y&=&120&&&&&& \\\\ 22y&-&y^2&=&120&&&&&& \\\\ \\\\ &&0&=&y^2&-&22y&+&120&& \\\\ &&0&=&y^2&-&12y&-&10y&+&120 \\\\ \\hline &&0&=&y(y&-&12)&-&10(y&-&12) \\\\ &&0&=&(y&-&12)(y&-&10)&& \\\\ \\\\ &&y&=&12,&10&&&&& \\end{array}[\/latex]
      \n[latex]\\therefore \\text{ numbers are }10, 12[\/latex]<\/li>\n
    2. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrccrrrrrrrrl} &&&&x&-&y&=&4&\\Rightarrow &x&=&y&+&4 \\\\ &&&&x&\\cdot &y&=&140&&&&&& \\\\ \\\\ &&&&(y&+&4)y&=&140&&&&&& \\\\ &&&&y^2&+&4y&=&140&&&&&& \\\\ \\\\ &&y^2&+&4y&-&140&=&0&&&&&& \\\\ y^2&-&10y&+&14y&-&140&=&0&&&&&& \\\\ \\hline y(y&-&10)&+&14(y&-&10)&=&0&&&&&& \\\\ &&(y&-&10)(y&+&14)&=&0&&&&&& \\\\ \\\\ &&&&&&y&=&10,&-14&&&&& \\\\ \\\\ &&&&&&y&=&10,&x&=&10&+&4&= 14 \\\\ &&&&&&y&=&-14,&x&=&-14&+&4&= -10 \\\\ \\end{array}[\/latex]
      \n[latex]\\therefore \\text{ numbers are }10, 14\\text{ and }-10, -14[\/latex]<\/li>\n
    3. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrcrrrrrrrrrl} &&&&A&-&B&=&8&\\Rightarrow &A&=&B&+&8 \\\\ &&&&A^2&+&B^2&=&320&&&&&& \\\\ \\\\ &&(B&+&8)^2&+&B^2&=&320&&&&&& \\\\ B^2&+&16B&+&64&+&B^2&=&320&&&&&& \\\\ &&&-&320&&&&-320&&&&&& \\\\ \\hline &&2B^2&+&16B&-&256&=&0&&&&&& \\\\ &&2(B^2&+&8B&-&128)&=&0&&&&&& \\\\ &&2(B&+&16)(B&-&8)&=&0&&&&&& \\\\ \\\\ &&&&&&B&=&-16,&8&&&&& \\\\ \\\\ &&&&&&\\therefore A&=&B&+&8&&&& \\\\ &&&&&&A&=&-8,&16&&&&& \\end{array}[\/latex]
      \n[latex]\\therefore (-16, -8)\\text{ and }(8,16)[\/latex]<\/li>\n
    4. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrcrrrrrrrrr} x, &x&+&2&&&&&&&&&& \\\\ \\\\ &&x^2&+&(x&+&2)^2&=&244&&&&& \\\\ x^2&+&x^2&+&4x&+&4&=&244&&&&& \\\\ &&&&&&-244&&-244&&&&& \\\\ \\hline &&2x^2&+&4x&-&240&=&0&&&&& \\\\ &&2(x^2&+&2x&-&120)&=&0&&&&& \\\\ &&2(x&-&10)(x&+&12)&=&0&&&&& \\\\ \\\\ &&&&&&x&=&10, &-12&&&& \\\\ \\\\ &&&&&&x&=&10, &x&+&2&=&12 \\\\ &&&&&&x&=&-12, &x&+&2&=&-10\\\\ \\end{array}[\/latex]
      \n[latex]\\therefore \\text{ numbers are }10, 12\\text{ or }-12, -10[\/latex]<\/li>\n
    5. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrrrrr} x,&x&+&2&&&&&&&& \\\\ \\\\ &&x^2&-&(x&+&2)^2&=&60&&& \\\\ x^2&-&(x^2&+&4x&+&4)&=&60&&& \\\\ x^2&-&x^2&-&4x&-&4&=&60&&& \\\\ &&&&&+&4&&+4&&& \\\\ \\hline &&&&&&\\dfrac{-4x}{-4}&=&\\dfrac{64}{-4}&&& \\\\ \\\\ &&&&&&x&=&-16&&& \\\\ \\\\ &&&&x&+&2&\\Rightarrow &-16&+&2& \\\\ &&&&&&&\\Rightarrow &-14&&& \\\\ \\end{array}[\/latex]
      \n[latex]-16, -14[\/latex]<\/li>\n
    6. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrcrrrl} x,&x&+&2&&&&& \\\\ \\\\ &&x^2&+&(x&+&2)^2&=&\\phantom{-}452 \\\\ x^2&+&x^2&+&4x&+&4&=&\\phantom{-}452 \\\\ &&&&&-&452&&-452 \\\\ \\hline &&2x^2&+&4x&-&448&=&0 \\\\ &&2(x^2&+&2x&-&224)&=&0 \\\\ &&2(x&-&14)(x&+&16)&=&0 \\\\ \\\\ &&&&&&x&=&14, -16 \\\\ \\\\ &&&&&&x&=&14 \\\\ &&&&x&+&2&=&16 \\\\ \\\\ &&&&&&x&=&-16 \\\\ &&&&x&+&2&=&-14 \\end{array}[\/latex]
      \n[latex]14,16\\text{ and }-16,-14[\/latex]<\/li>\n
    7. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrcrrrrrrrr} x,&x&+&2,&x&+&4&&&& \\\\ \\\\ &&x(x&+&2)&=&38&+&x&+&4 \\\\ x^2&+&2x&&&=&42&+&x&& \\\\ &-&x&-&42&&-42&-&x&& \\\\ \\hline x^2&+&x&-&42&=&0&&&& \\\\ (x&+&7)(x&-&6)&=&0&&&& \\\\ &&&&x&=&\\cancel{-7},&6&&& \\\\ \\end{array}[\/latex]
      \n[latex]\\therefore \\text{ numbers are }6,8,10[\/latex]<\/li>\n
    8. [latex]x, x+2, x+4[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrrrr} &&(x)(x&+&2)&=&52&+&x&+&4 \\\\ \\\\ x^2&+&2x&&&=&56&+&x&& \\\\ &-&x&-&56&&-56&-&x&& \\\\ \\hline x^2&+&x&-&56&=&0&&&& \\\\ (x&+&8)(x&-&7)&=&0&&&& \\\\ \\\\ &&&&x&=&\\cancel{-8}, 7&&&& \\end{array}[\/latex]
      \n[latex]\\therefore \\text{ numbers are }7,9,11[\/latex]<\/li>\n
    9. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrlrrr} &&A&=&T&+&4&&&&& \\\\ \\\\ A&\\cdot &T&=&80&+&(A&-&4)&(T&-&4) \\\\ (T&+&4)T&=&80&+&(T&+&\\cancel{4-4})&(T&-&4) \\\\ \\\\ T^2&+&4T&=&80&+&T^2&-&4T&&& \\\\ -T^2&+&4T&&&-&T^2&+&4T&&& \\\\ \\hline &&\\dfrac{8T}{8}&=&\\dfrac{80}{8}&&&&&&& \\\\ \\\\ &&T&=&10&&&&&&& \\\\ \\\\ &&\\therefore A&=&T&+&4&&&&& \\\\ &&A&=&10&+&4&=&14&&& \\end{array}[\/latex]<\/li>\n
    10. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrcrrrcrcrrrr} &&C&=&K&+&3&&&&&& \\\\ &&CK&=&(C&+&5)&&(K&+&5)&-&130 \\\\ \\\\ (K&+&3)K&=&(K&+&3&+&5)(K&+&5)&-&130 \\\\ K^2&+&3K&=&K^2&+&13K&+&40&-&130&& \\\\ -K^2&-&13K&&-K^2&-&13K&&&&&& \\\\ \\hline &&\\dfrac{-10K}{-10}&=&\\dfrac{-90}{-10}&&&&&&&& \\\\ \\\\ &&K&=&9&&&&&&&& \\\\ \\\\ &&\\therefore C&=&9&+&3&=&12&&&& \\end{array}[\/latex]<\/li>\n
    11. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrcrrrrrrrr} &&&&&&J&=&S&+&1&& \\\\ \\\\ &&(J&+&5)(S&+&5)&=&230&+&J&\\cdot &S \\\\ (S&+&1&+&5)(S&+&5)&=&230&+&(S&+&1)S \\\\ &&(S&+&6)(S&+&5)&=&230&+&S^2&+&S \\\\ \\\\ &&S^2&+&11S&+&30&=&S^2&+&S&+&230 \\\\ &&-S^2&-&S&-&30&&-S^2&-&S&-&30 \\\\ \\hline &&&&&&\\dfrac{10S}{10}&=&\\dfrac{200}{10}&&&& \\\\ \\\\ &&&&&&S&=&20&&&& \\\\ &&&&&&J&=&21&&&& \\end{array}[\/latex]<\/li>\n
    12. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrcrcrrrrrrrr} &&&&&&J&=&S&+&2&& \\\\ &&(S&+&2)(J&+&2)&=&48&+&S&\\cdot &J \\\\ \\\\ (S&+&2)(S&+&2&+&2)&=&48&+&S(S&+&2) \\\\ &&(S&+&2)(S&+&4)&=&48&+&S^2&+&25 \\\\ \\\\ &&S^2&+&6S&+&8&=&48&+&S^2&+&25 \\\\ &&-S^2&-&2S&-&8&&-8&-&S^2&-&25 \\\\ \\hline &&&&&&\\dfrac{4S}{4}&=&\\dfrac{40}{4}&&&& \\\\ \\\\ &&&&&&S&=&10&&&& \\\\ &&&&&&J&=&12&&&& \\end{array}[\/latex]<\/li>\n
    13. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{ll} \\begin{array}[t]{rrrrrrrrl} &&&&&&d&=&r\\cdot t \\\\ \\\\ &&&&r&\\cdot &t&=&240\\ \\\\ &&&&&&\\therefore r&=&\\dfrac{240}{t} \\\\ \\\\ &&(r&+&20)(t&-&1)&=&240 \\\\ &&(\\dfrac{240}{t}&+&20)(t&-&1)&=&240 \\\\ \\\\ \\cancel{240}&+&20t&-&\\dfrac{240}{t}&-&20&=&\\cancel{240} \\\\ \\\\ &&(20t&-&\\dfrac{240}{t}&-&20&=&0)(t) \\\\ \\\\ &&(20t^2&-&240&-&20t&=&0)(\\div 20) \\end{array} &\\hspace{0.25in} \\begin{array}[t]{rrcrcrl} t^2&-&12&-&t&=&0 \\\\ (t&-&4)(t&+&3)&=&0 \\\\ \\\\ &&&&t&=&4, \\cancel{-3} \\\\ \\\\ &&&&r&=&\\dfrac{240}{4}\\text{ or }60\\text{ km\/h} \\\\ \\\\ &&&&\\text{faster}&=&80\\text{ km\/h} \\end{array} \\end{array}[\/latex]<\/li>\n
    14. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{ll} \\begin{array}[t]{rrrrrrrrl} &&&&&&d&=&r\\cdot t \\\\ &&&&r&\\cdot &t&=&100 \\\\ &&&&&&r&=&\\dfrac{100}{t} \\\\ \\\\ &&(r&+&20)(t&-&0.5)&=&120 \\\\ &&(\\dfrac{100}{t}&+&20)(t&-&0.5)&=&120 \\\\ 100&+&20t&-&\\dfrac{50}{t}&-&10&=&120 \\\\ &&&&&-&120&&-120 \\\\ \\hline &&20t&-&30&-&\\dfrac{50}{t}&=&0 \\\\ \\\\ &&(20t&-&30&-&\\dfrac{50}{t}&=&0)(t) \\\\ \\\\ &&(20t^2&-&30t&-&50&=&0)(\\div 10) \\end{array} &\\hspace{0.25in} \\begin{array}[t]{rrrrrrl} 2t^2&-&3t&-&5&=&0 \\\\ &&&&t&=&\\dfrac{-(-3)\\pm \\sqrt{(-3)^2-4(2)(-5)}}{2(2)} \\\\ \\\\ &&&&t&=&\\dfrac{3\\pm 7}{4}=\\dfrac{10}{4}\\text{ or }\\cancel{\\dfrac{-4}{4}} \\\\ \\\\ &&&&t&=&2.5\\text{ h} \\end{array} \\end{array}[\/latex]
      \n[latex]\\text{Answer: }\\dfrac{100\\text{ km}}{2.5\\text{ h}}=\\dfrac{40\\text{ km}}{\\text{h}}, \\dfrac{120\\text{ km}}{2\\text{ h}}=\\dfrac{60\\text{ km}}{\\text{h}}[\/latex]<\/li>\n
    15. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{ll} \\begin{array}[t]{rrrrrrrrl} &&&&&&d&=&r\\cdot t \\\\ &&&&r&\\cdot &t&=&150 \\\\ &&&&&&r&=&\\dfrac{150}{t} \\\\ \\\\ &&(r&+&5)(t&-&1.5)&=&150 \\\\ &&(\\dfrac{150}{t}&+&5)(t&-&1.5)&=&150 \\\\ \\\\ \\cancel{150}&+&5t&-&\\dfrac{225}{t}&-&7.5&=&\\cancel{150} \\\\ \\\\ &&(5t&-&\\dfrac{225}{t}&-&7.5&=&0)(t) \\\\ \\\\ &&(5t^2&-&7.5t&-&225&=&0)(2) \\\\ &&(10t^2&-&15t&-&450&=&0)(\\div 5) \\end{array} &\\hspace{0.25in} \\begin{array}[t]{rrrrrrl} 2t^2&-&3t&-&90&=&0 \\\\ (t&+&6)(2t&-&15)&=&0 \\\\ &&&&t&=&\\cancel{-6}, \\dfrac{15}{2} \\\\ \\\\ &&&&r&=&\\dfrac{150}{t} \\\\ \\\\ &&&&r&=&\\dfrac{150}{\\dfrac{15}{2}} \\\\ \\\\ &&&&r&=&\\dfrac{150}{1}\\cdot \\dfrac{2}{15} \\\\ \\\\ &&&&r&=&20\\text{ km\/h} \\end{array} \\end{array}[\/latex]<\/li>\n
    16. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrl} &&&&&&d&=&r\\cdot t \\\\ &&&&r&\\cdot &t&=&180\\Rightarrow r=\\dfrac{180}{t} \\\\ \\\\ &&(r&+&15)(t&-&1)&=&180 \\\\ &&(\\dfrac{180}{t}&+&15)(t&-&1)&=&180 \\\\ \\\\ \\cancel{180}&+&15t&-&\\dfrac{180}{t}&-&15&=&\\cancel{180} \\\\ &&(15t&-&15&-&\\dfrac{180}{t}&=&0)(t) \\\\ &&(15t^2&-&15t&-&180&=&0)(\\div 15) \\\\ \\\\ &&t^2&-&t&-&12&=&0 \\\\ &&(t&-&4)(t&+&3)&=&0 \\\\ &&&&&&t&=&4, \\cancel{-3} \\\\ \\\\ &&&&&&r&=&\\dfrac{180}{4}=45 \\end{array}[\/latex]<\/li>\n
    17. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrcrrrl} &&&&r&\\cdot &t&=&72\\Rightarrow r=\\dfrac{72}{t} \\\\ \\\\ &&(r&+&12)(9&-&t)&=&72 \\\\ \\\\ &&(\\dfrac{72}{t}&+&12)(9&-&t)&=&72 \\\\ \\dfrac{648}{t}&+&108&-&72&-&12t&=&72 \\\\ &&&-&72&&&&-72 \\\\ \\hline &&(-12t&-&36&+&\\dfrac{648}{t}&=&0)(t) \\\\ \\\\ &&(-12t^2&-&36t&+&648&=&0)(\\div -12) \\\\ \\\\ &&t^2&+&3t&-&54&=&0 \\\\ &&(t&+&9)(t&-&6)&=&0 \\\\ &&&&&&t&=&\\cancel{-9}, 6 \\\\ \\\\ &&&&&&r&=&\\dfrac{72}{6}=12\\text{ (there)} \\\\ \\\\ &&&&&&r&=&24\\text{ (return)} \\end{array}[\/latex]<\/li>\n
    18. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrcrrrl} &&&&r&\\cdot &t&=&120\\Rightarrow r=\\dfrac{120}{t} \\\\ &&(r&+&10)(7&-&t)&=&120 \\\\ \\\\ &&(\\dfrac{120}{t}&+&10)(7&-&t)&=&120 \\\\ \\dfrac{840}{t}&+&70&-&120&-&10t&=&120 \\\\ &&&-&120&&&&-120 \\\\ \\hline &&(-10t&-&170&+&\\dfrac{840}{t}&=&0)(t) \\\\ &&(-10t^2&-&170t&+&840&=&0)(\\div -10) \\\\ \\\\ &&t^2&+&17t&-&84&=&0 \\\\ &&(t&+&21)(t&-&4)&=&0 \\\\ &&&&&&t&=&\\cancel{-21}, 4 \\\\ \\\\ &&&&&&r&=&\\dfrac{120}{4}\\text{ or }30\\text{ km\/h} \\\\ \\\\ &&&&r&+&10&=&40\\text{ km\/h} \\\\ \\end{array}[\/latex]<\/li>\n
    19. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrcrrrl} &&&&r&\\cdot &t&=&240\\Rightarrow r=\\dfrac{240}{t} \\\\ \\\\ &&(r&+&20)(t&-&1)&=&240 \\\\ &&(\\dfrac{240}{t}&+&20)(t&-&1)&=&240 \\\\ \\cancel{240}&+&20t&-&\\dfrac{240}{t}&-&20&=&\\cancel{240} \\\\ \\\\ &&(20t&-&20&-&\\dfrac{240}{t}&=&0)(t) \\\\ &&(20t^2&-&20t&-&240&=&0)(\\div 20) \\\\ \\\\ &&t^2&-&t&-&12&=&0 \\\\ &&(t&-&4)(t&+&3)&=&0 \\\\ &&&&&&t&=&4, \\cancel{-3} \\\\ \\\\ &&&&&&r&=&\\dfrac{240}{4}\\text{ or }60\\text{ km\/h} \\end{array}[\/latex]<\/li>\n
    20. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrcrrrl} &&&&r&\\cdot &t&=&600\\Rightarrow r=\\dfrac{600}{t} \\\\ \\\\ &&(r&-&50)(7&-&t)&=&600 \\\\ &&(\\dfrac{600}{t}&-&50)(7&-&t)&=&600 \\\\ \\dfrac{4200}{t}&-&350&-&600&+&50t&=&600 \\\\ &&&-&600&&&&-600 \\\\ \\hline &&(50t&-&1550&+&\\dfrac{4200}{t}&=&0)(t) \\\\ &&(50t^2&-&1550t&+&4200&=&0)(\\div 50) \\\\ \\\\ &&t^2&-&31t&+&84&=&0 \\\\ &&(t&-&3)(t&-&28)&=&0 \\\\ &&&&&&t&=&3, \\cancel{28} \\\\ \\\\ &&&&&&r&=&\\dfrac{600}{3}\\text{ or }200\\text{ km\/h} \\end{array}[\/latex]<\/li>\n
    21. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrlrrrl} L&=&4&+&W&&&& \\\\ \\text{Area}&=&L&\\cdot &W&&&& \\\\ \\\\ 60&=&(4&+&W)W&&&& \\\\ 60&=&4W&+&W^2&&&& \\\\ \\\\ 0&=&W^2&+&4W&-&60&& \\\\ 0&=&W^2&+&10W&-&6W&-&60 \\\\ \\hline 0&=&W(W&+&10)&-&6(W&+&10) \\\\ 0&=&(W&+&10)(W&-&6)&& \\\\ \\\\ W&=&\\cancel{-10},&6&&&&& \\\\ L&=&6&+&4&=&10&& \\end{array}[\/latex]<\/li>\n
    22. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrcrrrr} W&=&L&-&10&&&& \\\\ \\text{Area}&=&L&\\cdot &W&&&& \\\\ \\\\ 200&=&L(L&-&10)&&&& \\\\ 200&=&L^2&-&10L&&&& \\\\ \\\\ 0&=&L^2&-&10L&-&200&& \\\\ 0&=&L^2&+&10L&-&20L&-&200 \\\\ \\hline 0&=&L(L&+&10)&-&20(L&+&10) \\\\ 0&=&(L&+&10)(L&-&20)&& \\\\ \\\\ L&=&\\cancel{-10},&20&&&&& \\\\ W&=&20&-&10&=&10&& \\end{array}[\/latex]<\/li>\n
    23. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrcrcrcrl} &&&&&&\\text{Area}_{\\text{large}}&-&\\text{Area}_{\\text{small}}&=&2800\\text{ m}^2 \\\\ \\\\ &&(150&+&2x)(120&+&2x)&-&(150)(120)&=&2800 \\\\ \\cancel{18000}&+&240x&+&300x&+&4x^2&-&\\cancel{18000}&=&2800 \\\\ &&&&&&&-&2800&&-2800 \\\\ \\hline &&&&4x^2&+&540x&-&2800&=&0 \\\\ &&&&x^2&+&135x&-&700&=&0 \\\\ &&&&(x&-&5)(x&+&140)&=&0 \\\\ &&&&&&&&x&=&5, \\cancel{-140} \\end{array}[\/latex]
      \n[latex]\\text{walkway}=5\\text{ m}[\/latex]<\/li>\n
    24. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrcrcrcrl} &&&&&&\\text{Area}_{\\text{large}}&-&\\text{Area}_{\\text{small}}&=&74\\text{ m}^2 \\\\ \\\\ &&(25&+&2x)(10&+&2x)&-&(25)(10)&=&74 \\\\ \\cancel{250}&+&20x&+&50x&+&4x^2&-&\\cancel{250}&=&74 \\\\ &&&&&&&-&74&&-74 \\\\ \\hline &&&&4x^2&+&70x&-&74&=&0 \\\\ &&&&2x^2&+&35x&-&37&=&0 \\\\ &&&&(x&-&1)(2x&+&37)&=&0 \\\\ &&&&&&&&x&=&1, \\cancel{-\\dfrac{37}{2}} \\\\ \\end{array}[\/latex]
      \n[latex]\\text{the overlap}=1\\text{ m}[\/latex]<\/li>\n
    25. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrl} &&&&L&=&W&+&4 \\\\ &&L&\\cdot &W&=&60&& \\\\ \\\\ &&(W&+&4)W&=&60&& \\\\ W^2&+&4W&&&=&60&& \\\\ &&&-&60&&-60&& \\\\ \\hline W^2&+&4W&-&60&=&0&& \\\\ (W&-&6)(W&+&10)&=&0&& \\\\ \\\\ &&&&W&=&6,&\\cancel{-10}& \\\\ &&&&L&=&6&+&4=10 \\end{array}[\/latex]<\/li>\n
    26. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrcrr} &&(x&+&5)^2&=&4(x)^2&&&& \\\\ \\\\ x^2&+&10x&+&25&=&4x^2&&&& \\\\ -x^2&-&10x&-&25&&-x^2&-&10x&-&25 \\\\ \\hline &&&&0&=&3x^2&-&10x&-&25 \\\\ &&&&0&=&(x&-&5)(3x&+&5) \\\\ &&&&x&=&5, &\\cancel{-\\dfrac{5}{3}}&&& \\end{array}[\/latex]<\/li>\n
    27. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrlll} &&&&L&=&20&+&W \\\\ &&L&\\cdot &W&=&2400&& \\\\ \\\\ &&(20&+&W)W&=&2400&& \\\\ W^2&+&20W&&&=&2400&& \\\\ &&&-&2400&&-2400&& \\\\ \\hline W^2&+&20W&-&2400&=&0&& \\\\ (W&+&60)(W&-&40)&=&0&& \\\\ \\\\ &&&&W&=&\\cancel{-60},&40& \\\\ &&&&L&=&20&+&40=60 \\end{array}[\/latex]<\/li>\n
    28. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrcrrrrrrrr} &&&&&&L&=&W&+&8&& \\\\ &&(L&+&2)(W&+&2)&=&L&\\cdot &W&+&60 \\\\ \\\\ (W&+&8&+&2)(W&+&2)&=&(W&+&8)W&+&60 \\\\ &&W^2&+&12W&+&20&=&W^2&+&8W&+&60 \\\\ &&-W^2&-&8W&-&20&&-W^2&-&8W&-&20 \\\\ \\hline &&&&&&\\dfrac{4W}{4}&=&\\dfrac{40}{4}&&&& \\\\ \\\\ &&&&&&W&=&10&&&& \\\\ &&&&&&L&=&10&+&8&=&18 \\end{array}[\/latex]<\/li>\n<\/ol>\n","protected":false},"author":90,"menu_order":99,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"cc-by-nc-sa"},"back-matter-type":[],"contributor":[],"license":[56],"class_list":["post-2000","back-matter","type-back-matter","status-publish","hentry","license-cc-by-nc-sa"],"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/2000","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter"}],"about":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/types\/back-matter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/users\/90"}],"version-history":[{"count":1,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/2000\/revisions"}],"predecessor-version":[{"id":2001,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/2000\/revisions\/2001"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/2000\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/media?parent=2000"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter-type?post=2000"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/contributor?post=2000"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/license?post=2000"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}