2.10

<body topmargin=0 leftmargin=0 marginheight=0 marginwidth=0> <table cellpadding=0 cellspacing=0 border=0><tr> <td valign="top" colspan=3 height=24 BGCOLOR="#006666" TEXT="#080000"> <div> <FONT SIZE="4" COLOR="#FFFFFF" FACE="Georgia" ><B>M</B></FONT><FONT SIZE="4" COLOR="#FFFFFF" FACE="Georgia" ><B>atemáticas de la U</B></FONT><B> (MatU)  | </B><A HREF="http://www.jcbmat.com/" TARGET="_top" TITLE="http://www.jcbmat.com/"><U><B>Home</B></U></A><B> | </B><A HREF="index.htm" TARGET="_top" TITLE="Cálculo 21"><U><B>Cálculo de una variable</B></U></A><B> | </B><A HREF="id488.htm" TARGET="_top" TITLE="Leithold EC7"><U><B>Leithold</B></U></A><B> |</B></div> <div> </div> </td> </tr><tr> <td valign="top" rowspan="2" width=165 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B> <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:160px;height:600px" data-ad-client="ca-pub-1578463498418701" data-ad-slot="3073775660"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </B></div> </td> <td valign="top" height=432 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B>Ejercicios </B><B>2.10</B></div> <div> <B>Tasas de variación relacionadas</B></div> <bR> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:block" data-ad-client="ca-pub-1578463498418701" data-ad-slot="3926688482" data-ad-format="auto" data-full-width-responsive="true"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <bR> <div> <B>Enunciados y soluciones de los 45 problemas del Ejercicio 2.10</B></div> <bR> <div> <FONT SIZE="4" COLOR="#0033CC" FACE="Bell MT" ><B>E</B></FONT>n los ejercicio <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >1</FONT> a <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >8</FONT>, <I>x</I> y <I>y</I> son funciones de la tercera variable <I>t</I>.</div> <div> <IMG SRC="298263300.gif" border=0 width="355" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#1" TITLE="2.10"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/hgwaH81_QGE?t=30s" TARGET="_top" TITLE="https://youtu.be/hgwaH81_QGE?t=30s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          <IMG SRC="298450340.gif" border=0 width="336" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#2" TITLE="2.10"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/hgwaH81_QGE?t=3m5s" TARGET="_top" TITLE="https://youtu.be/hgwaH81_QGE?t=3m5s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> </div> <div> <IMG SRC="2985df300.gif" border=0 width="479" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#3" TITLE="2.10"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/hgwaH81_QGE?t=6m39s" TARGET="_top" TITLE="https://youtu.be/hgwaH81_QGE?t=6m39s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> </div> <div> <IMG SRC="29866b340.gif" border=0 width="619" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#4" TITLE="2.10"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/hgwaH81_QGE?t=9m2s" TARGET="_top" TITLE="https://youtu.be/hgwaH81_QGE?t=9m2s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> </div> <div> <IMG SRC="298d25340.gif" border=0 width="549" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#5" TITLE="2.10"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/_nzA0Q4-nQ4?t=14s" TARGET="_top" TITLE="https://youtu.be/_nzA0Q4-nQ4?t=14s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> </div> <div> <IMG SRC="2990ef300.gif" border=0 width="495" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#6" TITLE="2.10"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/_nzA0Q4-nQ4?t=6m8s" TARGET="_top" TITLE="https://youtu.be/_nzA0Q4-nQ4?t=6m8s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> </div> <div> <IMG SRC="2992e0300.gif" border=0 width="480" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#7" TITLE="2.10"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/_nzA0Q4-nQ4?t=10m21s" TARGET="_top" TITLE="https://youtu.be/_nzA0Q4-nQ4?t=10m21s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> </div> <div> <IMG SRC="29941d300.gif" border=0 width="541" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#8" TITLE="2.10"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/_nzA0Q4-nQ4?t=15m16s" TARGET="_top" TITLE="https://youtu.be/_nzA0Q4-nQ4?t=15m16s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>S</B></FONT>olución en video de los Ejercicios <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >1</FONT> a <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >4</FONT> <A HREF="https://youtu.be/hgwaH81_QGE" TARGET="_top" TITLE="https://youtu.be/hgwaH81_QGE"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>S</B></FONT>olución en video de los Ejercicios <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >5</FONT> a <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >8</FONT> <A HREF="https://youtu.be/_nzA0Q4-nQ4" TARGET="_top" TITLE="https://youtu.be/_nzA0Q4-nQ4"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <div> <FONT SIZE="4" COLOR="#0033CC" FACE="Bell MT" ><B>E</B></FONT>n los problemas de tasas de variación relacionadas de los ejercicios siguientes, defina precisamente todas las variables como cantidades (números y unidades de medición). Utilice la variable <I>t</I> para representar el tiempo y defina las otras variables de modo que dependan de <I>t</I>. Asegúrese de escribir una conclusión.</div> <div> <FONT SIZE="4" COLOR="#CC0033" FACE="Times New Roman" >9. </FONT> Un niño vuela una cometa a una altura de 40 pies, y lo hace moviéndose horizontalmente a una tasa de de 3 pie/s. Si la cuerda está tensa, ¿a qué tasa se afloja cuando la longitud de la cuerda suelta es de 50 pie?  <A HREF="#8" TITLE="2.10"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/_nzA0Q4-nQ4?t=15m16s" TARGET="_top" TITLE="https://youtu.be/_nzA0Q4-nQ4?t=15m16s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <FONT SIZE="4" COLOR="#CC0033" FACE="Times New Roman" >10. </FONT> Se infla un globo esférico de modo que su volumen se incrementa a una tasa de 5 m^3/min. ¿A qué tasa aumenta el diámetro cuando este es de 12 m?   <A HREF="#10" TITLE="2.10"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/_nzA0Q4-nQ4?t=15m16s" TARGET="_top" TITLE="https://youtu.be/_nzA0Q4-nQ4?t=15m16s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <bR> <DIV ALIGN="LEFT"></DIV> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:200px;height:90px" data-ad-client="ca-pub-1578463498418701" data-ad-slot="4434969515"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>         <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:200px;height:90px" data-ad-client="ca-pub-1578463498418701" data-ad-slot="4434969515"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <div>  <A NAME="1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1</div> <div> <IMG SRC="2989e7b10.gif" border=0 width="743" height="177" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>2</div> <div> <IMG SRC="298a8fcf0.gif" border=0 width="911" height="207" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>3</div> <div> <IMG SRC="298be0830.gif" border=0 width="736" height="131" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="4"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>4</div> <div> <IMG SRC="298c84c10.gif" border=0 width="900" height="193" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="5"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>5</div> <div>  <IMG SRC="298e54640.gif" border=0 width="852" height="356" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="6"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>6</div> <div> <IMG SRC="298fd3b40.gif" border=0 width="723" height="180" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="7"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>7</div> <div> <IMG SRC="29911cc10.gif" border=0 width="796" height="193" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="8"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>8</div> <div> <IMG SRC="2993d0b40.gif" border=0 width="720" height="180" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="9"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>9</div> <div> 9. <FONT SIZE="4" COLOR="#111111" FACE="Bell MT" > Un niño vuela una cometa a una altura de 40 pies, y lo hace moviéndose horizontalmente a una tasa de de 3 pie/s. Si la cuerda está tensa, ¿a qué tasa se afloja cuando la longitud de la cuerda suelta es de 50 pie?</FONT></div> <div> <IMG SRC="299754c10.gif" border=0 width="852" height="449" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="10"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>10</div> <div> 10. <FONT SIZE="4" COLOR="#111111" FACE="Bell MT" > Se infla un globo esférico de modo que su volumen se incrementa a una tasa de 5 m^3/min. ¿A qué tasa aumenta el diámetro cuando éste es de 12 m?</FONT></div> <div> Solución -<FONT SIZE="4" COLOR="#339900" FACE="Bell MT" > Juan Beltrán</FONT>:</div> <div> <IMG SRC="29994de10.gif" border=0 width="845" height="481" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="11"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>11</div> <div> <IMG SRC="263ea91c0.gif" border=0 width="937" height="796" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div>  <A NAME="27"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>27</div> <div>  <iframe width="420" height="315" src="//www.youtube.com/embed/S76gxSCMr60" frameborder="0" allowfullscreen></iframe>  </div> <bR> <div> <B> <img src="HTMLobj-11792/aniGif.gif" border="0" align="bottom" width="550" height="14">  </B></div> <bR> <div> <IMG SRC="2c128c8c0.gif" border=0 width="140" height="140" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <A HREF="http://twitter.com/Calculo21" TARGET="_top" TITLE="http://twitter.com/Calculo21"><U><IMG SRC="248328280.jpg" border=0 width="40" height="40" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><A HREF="http://www.facebook.com/juan.beltran.58511" TARGET="_top" TITLE="http://www.facebook.com/juan.beltran.585"><U><IMG SRC="248428280.gif" border=0 width="40" height="40" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><A HREF="https://www.youtube.com/user/MatematicaAplicadas" TARGET="_top" TITLE="https://www.youtube.com/user/MatematicaA"><U><IMG SRC="248528280.jpg" border=0 width="40" height="40" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <div>  <a href="http://twitter.com/Calculo21" class="twitter-follow-button" data-button="grey" data-text-color="#FFFFFF" data-link-color="#00AEFF" data-show-count="false" data-lang="es">Follow @Calculo21</a> <script src="http://platform.twitter.com/widgets.js" type="text/javascript"></script>  </div> <bR> <div> Por: <FONT SIZE="4" COLOR="#FF9900" FACE="Times New Roman" >Juan Carlos </FONT><FONT SIZE="4" COLOR="#0000FF" FACE="Times New Roman" >Beltrán</FONT> <FONT SIZE="4" COLOR="#FF0000" FACE="Times New Roman" >Beltrán </FONT></div> <bR> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:970px;height:250px" data-ad-client="ca-pub-6266400581769473" data-ad-slot="7846235711"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); 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