{"id":2010,"date":"2021-12-02T19:40:27","date_gmt":"2021-12-03T00:40:27","guid":{"rendered":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/midterm-3-version-c\/"},"modified":"2022-11-02T10:39:06","modified_gmt":"2022-11-02T14:39:06","slug":"midterm-3-version-c","status":"publish","type":"back-matter","link":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/midterm-3-version-c\/","title":{"raw":"Midterm 3: Version C Answer Key","rendered":"Midterm 3: Version C Answer Key"},"content":{"raw":"
    \n \t
  1. [latex]\\dfrac{\\cancel{15}3\\cancel{m^3}}{4\\cancel{n^2}}\\cdot \\dfrac{\\cancel{17}\\cancel{n^3}}{\\cancel{30}\\cancel{10}2\\cancel{m^3}}\\cdot \\dfrac{\\cancel{3}1m^4}{\\cancel{34}2n^{\\cancel{2}}}\\Rightarrow \\dfrac{3m^4}{16n}[\/latex]<\/li>\n \t
  2. [latex]\\dfrac{5v^2-25v}{5v+25}\\cdot \\dfrac{10v}{v^2-11v+30}\\Rightarrow \\dfrac{\\cancel{5}v\\cancel{(v-5)}}{\\cancel{5}(v+5)}\\cdot \\dfrac{10v}{\\cancel{(v-5)}(v-6)}\\Rightarrow \\dfrac{10v^2}{(v+5)(v-6)}[\/latex]<\/li>\n \t
  3. [latex]\\left(\\dfrac{8}{2x}=\\dfrac{2}{x}+1\\right)(2x) [\/latex]\n[latex]\\begin{array}{rrl}\n8&=&2\\cdot 2+1(2x) \\\\\n8&=&\\phantom{-}4+2x \\\\\n-4&&-4 \\\\\n\\hline\n\\dfrac{4}{2}&=&\\dfrac{2x}{2} \\\\ \\\\\nx&=&2\n\\end{array}[\/latex]<\/li>\n \t
  4. [latex]\\begin{array}[t]{l}\n\\dfrac{\\left(\\dfrac{x^2}{y^2}-16\\right)y^3}{\\left(\\dfrac{x+4y}{y^3}\\right)y^3}\\Rightarrow \\dfrac{x^2y-16y^3}{x+4y}\\Rightarrow \\dfrac{y(x^2-16y^2)}{x+4y}\\Rightarrow\\dfrac{y(x-4y)\\cancel{(x+4y)}}{\\cancel{x+4y}} \\\\ \\\\\n\\Rightarrow y(x-4y)\n\\end{array}[\/latex]<\/li>\n \t
  5. [latex]5y+2\\cdot 9y+6y\\sqrt{y}[\/latex]\n[latex]\\begin{array}[t]{l}\n23y+6y\\sqrt{y}\n\\end{array}[\/latex]<\/li>\n \t
  6. [latex]\\dfrac{(28)(7+3\\sqrt{5})}{(7-3\\sqrt{5})(7+3\\sqrt{5})}\\Rightarrow \\dfrac{196+84\\sqrt{5}}{49-9\\cdot 5}\\Rightarrow \\dfrac{196+84\\sqrt{5}}{4}\\Rightarrow 49+21\\sqrt{5}[\/latex]<\/li>\n \t
  7. [latex](27a^{-\\frac{3}{8}})^{\\frac{1}{3}}[\/latex]\n[latex]\\begin{array}[t]{l}\n27^{\\frac{1}{3}}a^{-\\frac{3}{8}\\cdot \\frac{1}{3}} \\\\ \\\\\n3a^{-\\frac{1}{8}} \\\\ \\\\\n\\dfrac{3}{a^{\\frac{1}{8}}}\\Rightarrow \\dfrac{3}{\\sqrt[8]{a}}\n\\end{array}[\/latex]<\/li>\n \t
  8. [latex](\\sqrt{3x-2})^2=(\\sqrt{5x+4})^2 [\/latex]\n[latex]\\begin{array}{rrrrrrrr}\n&3x&-&2&=&5x&+&4 \\\\\n-&3x&-&4&&-3x&-&4 \\\\\n\\hline\n&&&-6&=&2x&& \\\\ \\\\\n&&&x&=&\\dfrac{-6}{2}&=&-3\n\\end{array}[\/latex]<\/li>\n \t
  9. [latex]\\phantom{a}[\/latex]\n
      \n \t
    1. [latex]\\begin{array}{rrl}\\\\\n\\dfrac{2x^2}{2}&=&\\dfrac{72}{2} \\\\ \\\\\nx^2&=&36 \\\\\nx&=&\\pm 6\n\\end{array}[\/latex]<\/li>\n \t
    2. [latex]\\begin{array}{rrl}\\\\\n2x^2-8x&=&0 \\\\\n2x(x-4)&=&0 \\\\\nx&=&0, 4\n\\end{array}[\/latex]<\/li>\n<\/ol>\n<\/li>\n \t
    3. [latex]\\phantom{a}[\/latex]\n
        \n \t
      1. [latex]\\begin{array}{rrl}\\\\\n(x+5)(x+1)&=&0 \\\\\nx&=&-1, -5\n\\end{array}[\/latex]<\/li>\n \t
      2. Quadratic:\n[latex]\\begin{array}[t]{l}\nx^2-10x+4=0 \\\\ \\\\\n\\dfrac{-(-10)\\pm \\sqrt{(-10)^2-4(1)(4)}}{2} \\\\ \\\\\n\\dfrac{-10\\pm \\sqrt{100-16}}{2} \\\\ \\\\\n\\dfrac{10\\pm \\sqrt{84}}{2} \\\\ \\\\\n\\dfrac{10\\pm 2\\sqrt{21}}{2}\\Rightarrow 5 \\pm \\sqrt{21}\n\\end{array}[\/latex]<\/li>\n<\/ol>\n<\/li>\n \t
      3. [latex]\\left(\\dfrac{8}{4x}=\\dfrac{2}{x}+3\\right)(4x)[\/latex]\n[latex]\\begin{array}{rrrrl}\n8&=&8&+&3(4x) \\\\\n-8&&-8&& \\\\\n\\hline\n\\dfrac{0}{12}&=&\\dfrac{12x}{12}&& \\\\ \\\\\nx&=&0&&\\therefore \\text{Undefined. No solution}\n\\end{array}[\/latex]<\/li>\n \t
      4. [latex]\\text{Let }u=x^2[\/latex]\n[latex]\\begin{array}[t]{rrl}\nu^2-17u+16&=&0 \\\\\n(u-16)(u-1)&=&0 \\\\ \\\\\n(x^2-16)(x^2-1)&=&0 \\\\\n(x-4)(x+4)(x-1)(x+1)&=&0 \\\\\nx&=& \\pm 1, \\pm 4\n\\end{array}[\/latex]<\/li>\n \t
      5. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{ll}\n\\begin{array}[t]{rrl}\nL&=&W+6 \\\\\n\\text{Area}&=&12+\\text{Perimeter} \\\\ \\\\\nL\\cdot W&=& 12+2L+2W \\\\\n(W+6)W&=&12+2(W+6)+2W \\\\\nW^2+6W&=&12+2W+12+2W\n\\end{array}\n& \\hspace{0.25in}\n\\begin{array}[t]{rrrrcrr}\n0&=&W^2&+&6W&& \\\\\n&&&-&4W&-&24 \\\\\n\\hline\n0&=&W^2&+&2W&-&24 \\\\\n0&=&(W&+&6)(W&-&4) \\\\\nW&=&\\cancel{-6},&4&&& \\\\ \\\\\nL&=&W&+&6&& \\\\\nL&=&4&+&6&& \\\\\nL&=&10&&&& \\\\\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
      6. [latex]x, x+2, x+4 [\/latex]\n[latex]\\begin{array}[t]{ll}\n\\begin{array}[t]{rrrrrrrrrrr}\n&&x(x&+&4)&=&31&+&x&+&2 \\\\\nx^2&+&4x&&&=&33&+&x&& \\\\\n&-&x&-&33&&-33&-&x&& \\\\\n\\hline\nx^2&+&3x&-&33&=&0&&&&\n\\end{array}\n& \\hspace{0.25in}\n\\begin{array}[t]{l}\n\\dfrac{-b\\pm \\sqrt{b^2-4ac}}{2a} \\\\ \\\\\n\\dfrac{-3\\pm \\sqrt{3^2-4(1)(-33)}}{2} \\\\ \\\\\n\\dfrac{-3\\pm \\sqrt{141}}{2}\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
      7. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrcl}\n&d&=&r&\\cdot &t \\\\\n\\text{To outpost:}&60&=&(B&-&C)5 \\\\\n\\text{Back:}&60&=&(B&+&C)3 \\\\ \\\\\n&60&=&5B&-&5C \\\\\n&60&=&3B&+&3C \\\\ \\\\\n&12&=&B&-&C \\\\\n+&20&=&B&+&C \\\\\n\\hline\n&32&=&2B&& \\\\\n&\\therefore B&=&16&\\text{ km\/h}& \\\\ \\\\\n&\\therefore B&+&C&=&\\phantom{-}20 \\\\\n&16&+&C&=&\\phantom{-}20 \\\\\n-&16&&&&-16 \\\\\n\\hline\n&&&C&=&4\\text{ km\/h}\n\\end{array}[\/latex]<\/li>\n<\/ol>","rendered":"
          \n
        1. [latex]\\dfrac{\\cancel{15}3\\cancel{m^3}}{4\\cancel{n^2}}\\cdot \\dfrac{\\cancel{17}\\cancel{n^3}}{\\cancel{30}\\cancel{10}2\\cancel{m^3}}\\cdot \\dfrac{\\cancel{3}1m^4}{\\cancel{34}2n^{\\cancel{2}}}\\Rightarrow \\dfrac{3m^4}{16n}[\/latex]<\/li>\n
        2. [latex]\\dfrac{5v^2-25v}{5v+25}\\cdot \\dfrac{10v}{v^2-11v+30}\\Rightarrow \\dfrac{\\cancel{5}v\\cancel{(v-5)}}{\\cancel{5}(v+5)}\\cdot \\dfrac{10v}{\\cancel{(v-5)}(v-6)}\\Rightarrow \\dfrac{10v^2}{(v+5)(v-6)}[\/latex]<\/li>\n
        3. [latex]\\left(\\dfrac{8}{2x}=\\dfrac{2}{x}+1\\right)(2x)[\/latex]
          \n[latex]\\begin{array}{rrl} 8&=&2\\cdot 2+1(2x) \\\\ 8&=&\\phantom{-}4+2x \\\\ -4&&-4 \\\\ \\hline \\dfrac{4}{2}&=&\\dfrac{2x}{2} \\\\ \\\\ x&=&2 \\end{array}[\/latex]<\/li>\n
        4. [latex]\\begin{array}[t]{l} \\dfrac{\\left(\\dfrac{x^2}{y^2}-16\\right)y^3}{\\left(\\dfrac{x+4y}{y^3}\\right)y^3}\\Rightarrow \\dfrac{x^2y-16y^3}{x+4y}\\Rightarrow \\dfrac{y(x^2-16y^2)}{x+4y}\\Rightarrow\\dfrac{y(x-4y)\\cancel{(x+4y)}}{\\cancel{x+4y}} \\\\ \\\\ \\Rightarrow y(x-4y) \\end{array}[\/latex]<\/li>\n
        5. [latex]5y+2\\cdot 9y+6y\\sqrt{y}[\/latex]
          \n[latex]\\begin{array}[t]{l} 23y+6y\\sqrt{y} \\end{array}[\/latex]<\/li>\n
        6. [latex]\\dfrac{(28)(7+3\\sqrt{5})}{(7-3\\sqrt{5})(7+3\\sqrt{5})}\\Rightarrow \\dfrac{196+84\\sqrt{5}}{49-9\\cdot 5}\\Rightarrow \\dfrac{196+84\\sqrt{5}}{4}\\Rightarrow 49+21\\sqrt{5}[\/latex]<\/li>\n
        7. [latex](27a^{-\\frac{3}{8}})^{\\frac{1}{3}}[\/latex]
          \n[latex]\\begin{array}[t]{l} 27^{\\frac{1}{3}}a^{-\\frac{3}{8}\\cdot \\frac{1}{3}} \\\\ \\\\ 3a^{-\\frac{1}{8}} \\\\ \\\\ \\dfrac{3}{a^{\\frac{1}{8}}}\\Rightarrow \\dfrac{3}{\\sqrt[8]{a}} \\end{array}[\/latex]<\/li>\n
        8. [latex](\\sqrt{3x-2})^2=(\\sqrt{5x+4})^2[\/latex]
          \n[latex]\\begin{array}{rrrrrrrr} &3x&-&2&=&5x&+&4 \\\\ -&3x&-&4&&-3x&-&4 \\\\ \\hline &&&-6&=&2x&& \\\\ \\\\ &&&x&=&\\dfrac{-6}{2}&=&-3 \\end{array}[\/latex]<\/li>\n
        9. [latex]\\phantom{a}[\/latex]\n
            \n
          1. [latex]\\begin{array}{rrl}\\\\ \\dfrac{2x^2}{2}&=&\\dfrac{72}{2} \\\\ \\\\ x^2&=&36 \\\\ x&=&\\pm 6 \\end{array}[\/latex]<\/li>\n
          2. [latex]\\begin{array}{rrl}\\\\ 2x^2-8x&=&0 \\\\ 2x(x-4)&=&0 \\\\ x&=&0, 4 \\end{array}[\/latex]<\/li>\n<\/ol>\n<\/li>\n
          3. [latex]\\phantom{a}[\/latex]\n
              \n
            1. [latex]\\begin{array}{rrl}\\\\ (x+5)(x+1)&=&0 \\\\ x&=&-1, -5 \\end{array}[\/latex]<\/li>\n
            2. Quadratic:
              \n[latex]\\begin{array}[t]{l} x^2-10x+4=0 \\\\ \\\\ \\dfrac{-(-10)\\pm \\sqrt{(-10)^2-4(1)(4)}}{2} \\\\ \\\\ \\dfrac{-10\\pm \\sqrt{100-16}}{2} \\\\ \\\\ \\dfrac{10\\pm \\sqrt{84}}{2} \\\\ \\\\ \\dfrac{10\\pm 2\\sqrt{21}}{2}\\Rightarrow 5 \\pm \\sqrt{21} \\end{array}[\/latex]<\/li>\n<\/ol>\n<\/li>\n
            3. [latex]\\left(\\dfrac{8}{4x}=\\dfrac{2}{x}+3\\right)(4x)[\/latex]
              \n[latex]\\begin{array}{rrrrl} 8&=&8&+&3(4x) \\\\ -8&&-8&& \\\\ \\hline \\dfrac{0}{12}&=&\\dfrac{12x}{12}&& \\\\ \\\\ x&=&0&&\\therefore \\text{Undefined. No solution} \\end{array}[\/latex]<\/li>\n
            4. [latex]\\text{Let }u=x^2[\/latex]
              \n[latex]\\begin{array}[t]{rrl} u^2-17u+16&=&0 \\\\ (u-16)(u-1)&=&0 \\\\ \\\\ (x^2-16)(x^2-1)&=&0 \\\\ (x-4)(x+4)(x-1)(x+1)&=&0 \\\\ x&=& \\pm 1, \\pm 4 \\end{array}[\/latex]<\/li>\n
            5. [latex]\\phantom{a}[\/latex]
              \n[latex]\\begin{array}[t]{ll} \\begin{array}[t]{rrl} L&=&W+6 \\\\ \\text{Area}&=&12+\\text{Perimeter} \\\\ \\\\ L\\cdot W&=& 12+2L+2W \\\\ (W+6)W&=&12+2(W+6)+2W \\\\ W^2+6W&=&12+2W+12+2W \\end{array} & \\hspace{0.25in} \\begin{array}[t]{rrrrcrr} 0&=&W^2&+&6W&& \\\\ &&&-&4W&-&24 \\\\ \\hline 0&=&W^2&+&2W&-&24 \\\\ 0&=&(W&+&6)(W&-&4) \\\\ W&=&\\cancel{-6},&4&&& \\\\ \\\\ L&=&W&+&6&& \\\\ L&=&4&+&6&& \\\\ L&=&10&&&& \\\\ \\end{array} \\end{array}[\/latex]<\/li>\n
            6. [latex]x, x+2, x+4[\/latex]
              \n[latex]\\begin{array}[t]{ll} \\begin{array}[t]{rrrrrrrrrrr} &&x(x&+&4)&=&31&+&x&+&2 \\\\ x^2&+&4x&&&=&33&+&x&& \\\\ &-&x&-&33&&-33&-&x&& \\\\ \\hline x^2&+&3x&-&33&=&0&&&& \\end{array} & \\hspace{0.25in} \\begin{array}[t]{l} \\dfrac{-b\\pm \\sqrt{b^2-4ac}}{2a} \\\\ \\\\ \\dfrac{-3\\pm \\sqrt{3^2-4(1)(-33)}}{2} \\\\ \\\\ \\dfrac{-3\\pm \\sqrt{141}}{2} \\end{array} \\end{array}[\/latex]<\/li>\n
            7. [latex]\\phantom{a}[\/latex]
              \n[latex]\\begin{array}[t]{rrrrcl} &d&=&r&\\cdot &t \\\\ \\text{To outpost:}&60&=&(B&-&C)5 \\\\ \\text{Back:}&60&=&(B&+&C)3 \\\\ \\\\ &60&=&5B&-&5C \\\\ &60&=&3B&+&3C \\\\ \\\\ &12&=&B&-&C \\\\ +&20&=&B&+&C \\\\ \\hline &32&=&2B&& \\\\ &\\therefore B&=&16&\\text{ km\/h}& \\\\ \\\\ &\\therefore B&+&C&=&\\phantom{-}20 \\\\ &16&+&C&=&\\phantom{-}20 \\\\ -&16&&&&-16 \\\\ \\hline &&&C&=&4\\text{ km\/h} \\end{array}[\/latex]<\/li>\n<\/ol>\n","protected":false},"author":90,"menu_order":104,"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-2010","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\/2010","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\/2010\/revisions"}],"predecessor-version":[{"id":2011,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/2010\/revisions\/2011"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/2010\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/media?parent=2010"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter-type?post=2010"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/contributor?post=2010"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/license?post=2010"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}