Related Rates Problem Involving Airplanes












4












$begingroup$


I took a test yesterday, and would like to know how to answer this specific question on the exam:



One airplane flew over an airport at the rate of $300$ mi/hr. Ten minutes later another airplane flew over the airport at $240$ mi/hr. If the first airplane was flying west and the second flying south (both at the same altitude), determine the rate at which they were separating $20$ minutes after the second plane flew over the airport.



I know that the pythagorean theorem should be used: $x^2 + y^2 = z^2$; but I don't know what to use for the $x, y$, and $z$ values.



$dy/dt = 240$ mi/hr



$dx/dt = 300$ mi/hr






calculus







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Let the position of the first plane be x and the position of the second be y. You're given dx/dt and dy/dt and you can solve for the initial x and y values. Then use the Pythagorean theorem to find the distance between the two planes and differentiate with respect to time.
    $endgroup$
    – Pranav Marathe
    Jan 29 '15 at 23:31






  • 1




    $begingroup$
    To attract answers to your question, please add some context and background information. For example, where did you encounter this problem (e.g. a book, class, real-life)? Please also show your attempt; seeing your work helps us help you. If this is homework, please read this post.
    $endgroup$
    – apnorton
    Jan 29 '15 at 23:46






  • 1




    $begingroup$
    @anorton Thank you for your help. I added some background information.
    $endgroup$
    – Audrey
    Jan 30 '15 at 0:03










  • $begingroup$
    @Audrey note that if one plane flies into west you can view that as a vector pointing to the left, that is in negative x-direction on a plane (-something,0). The plane that flies to the south, you can view as vector pointing down, or negative y-direction on a plane (0, -somethingelse). so you can apply pythagorean theorem something^2+somethingelse^2=result^2...
    $endgroup$
    – Loreno Heer
    Jan 30 '15 at 0:22
















4












$begingroup$


I took a test yesterday, and would like to know how to answer this specific question on the exam:



One airplane flew over an airport at the rate of $300$ mi/hr. Ten minutes later another airplane flew over the airport at $240$ mi/hr. If the first airplane was flying west and the second flying south (both at the same altitude), determine the rate at which they were separating $20$ minutes after the second plane flew over the airport.



I know that the pythagorean theorem should be used: $x^2 + y^2 = z^2$; but I don't know what to use for the $x, y$, and $z$ values.



$dy/dt = 240$ mi/hr



$dx/dt = 300$ mi/hr






calculus







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Let the position of the first plane be x and the position of the second be y. You're given dx/dt and dy/dt and you can solve for the initial x and y values. Then use the Pythagorean theorem to find the distance between the two planes and differentiate with respect to time.
    $endgroup$
    – Pranav Marathe
    Jan 29 '15 at 23:31






  • 1




    $begingroup$
    To attract answers to your question, please add some context and background information. For example, where did you encounter this problem (e.g. a book, class, real-life)? Please also show your attempt; seeing your work helps us help you. If this is homework, please read this post.
    $endgroup$
    – apnorton
    Jan 29 '15 at 23:46






  • 1




    $begingroup$
    @anorton Thank you for your help. I added some background information.
    $endgroup$
    – Audrey
    Jan 30 '15 at 0:03










  • $begingroup$
    @Audrey note that if one plane flies into west you can view that as a vector pointing to the left, that is in negative x-direction on a plane (-something,0). The plane that flies to the south, you can view as vector pointing down, or negative y-direction on a plane (0, -somethingelse). so you can apply pythagorean theorem something^2+somethingelse^2=result^2...
    $endgroup$
    – Loreno Heer
    Jan 30 '15 at 0:22














4












4








4


1



$begingroup$


I took a test yesterday, and would like to know how to answer this specific question on the exam:



One airplane flew over an airport at the rate of $300$ mi/hr. Ten minutes later another airplane flew over the airport at $240$ mi/hr. If the first airplane was flying west and the second flying south (both at the same altitude), determine the rate at which they were separating $20$ minutes after the second plane flew over the airport.



I know that the pythagorean theorem should be used: $x^2 + y^2 = z^2$; but I don't know what to use for the $x, y$, and $z$ values.



$dy/dt = 240$ mi/hr



$dx/dt = 300$ mi/hr






calculus







share|cite|improve this question











$endgroup$




I took a test yesterday, and would like to know how to answer this specific question on the exam:



One airplane flew over an airport at the rate of $300$ mi/hr. Ten minutes later another airplane flew over the airport at $240$ mi/hr. If the first airplane was flying west and the second flying south (both at the same altitude), determine the rate at which they were separating $20$ minutes after the second plane flew over the airport.



I know that the pythagorean theorem should be used: $x^2 + y^2 = z^2$; but I don't know what to use for the $x, y$, and $z$ values.



$dy/dt = 240$ mi/hr



$dx/dt = 300$ mi/hr






calculus




calculus






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 30 '15 at 0:03







Audrey

















asked Jan 29 '15 at 23:21









AudreyAudrey

448




448








  • 1




    $begingroup$
    Let the position of the first plane be x and the position of the second be y. You're given dx/dt and dy/dt and you can solve for the initial x and y values. Then use the Pythagorean theorem to find the distance between the two planes and differentiate with respect to time.
    $endgroup$
    – Pranav Marathe
    Jan 29 '15 at 23:31






  • 1




    $begingroup$
    To attract answers to your question, please add some context and background information. For example, where did you encounter this problem (e.g. a book, class, real-life)? Please also show your attempt; seeing your work helps us help you. If this is homework, please read this post.
    $endgroup$
    – apnorton
    Jan 29 '15 at 23:46






  • 1




    $begingroup$
    @anorton Thank you for your help. I added some background information.
    $endgroup$
    – Audrey
    Jan 30 '15 at 0:03










  • $begingroup$
    @Audrey note that if one plane flies into west you can view that as a vector pointing to the left, that is in negative x-direction on a plane (-something,0). The plane that flies to the south, you can view as vector pointing down, or negative y-direction on a plane (0, -somethingelse). so you can apply pythagorean theorem something^2+somethingelse^2=result^2...
    $endgroup$
    – Loreno Heer
    Jan 30 '15 at 0:22














  • 1




    $begingroup$
    Let the position of the first plane be x and the position of the second be y. You're given dx/dt and dy/dt and you can solve for the initial x and y values. Then use the Pythagorean theorem to find the distance between the two planes and differentiate with respect to time.
    $endgroup$
    – Pranav Marathe
    Jan 29 '15 at 23:31






  • 1




    $begingroup$
    To attract answers to your question, please add some context and background information. For example, where did you encounter this problem (e.g. a book, class, real-life)? Please also show your attempt; seeing your work helps us help you. If this is homework, please read this post.
    $endgroup$
    – apnorton
    Jan 29 '15 at 23:46






  • 1




    $begingroup$
    @anorton Thank you for your help. I added some background information.
    $endgroup$
    – Audrey
    Jan 30 '15 at 0:03










  • $begingroup$
    @Audrey note that if one plane flies into west you can view that as a vector pointing to the left, that is in negative x-direction on a plane (-something,0). The plane that flies to the south, you can view as vector pointing down, or negative y-direction on a plane (0, -somethingelse). so you can apply pythagorean theorem something^2+somethingelse^2=result^2...
    $endgroup$
    – Loreno Heer
    Jan 30 '15 at 0:22








1




1




$begingroup$
Let the position of the first plane be x and the position of the second be y. You're given dx/dt and dy/dt and you can solve for the initial x and y values. Then use the Pythagorean theorem to find the distance between the two planes and differentiate with respect to time.
$endgroup$
– Pranav Marathe
Jan 29 '15 at 23:31




$begingroup$
Let the position of the first plane be x and the position of the second be y. You're given dx/dt and dy/dt and you can solve for the initial x and y values. Then use the Pythagorean theorem to find the distance between the two planes and differentiate with respect to time.
$endgroup$
– Pranav Marathe
Jan 29 '15 at 23:31




1




1




$begingroup$
To attract answers to your question, please add some context and background information. For example, where did you encounter this problem (e.g. a book, class, real-life)? Please also show your attempt; seeing your work helps us help you. If this is homework, please read this post.
$endgroup$
– apnorton
Jan 29 '15 at 23:46




$begingroup$
To attract answers to your question, please add some context and background information. For example, where did you encounter this problem (e.g. a book, class, real-life)? Please also show your attempt; seeing your work helps us help you. If this is homework, please read this post.
$endgroup$
– apnorton
Jan 29 '15 at 23:46




1




1




$begingroup$
@anorton Thank you for your help. I added some background information.
$endgroup$
– Audrey
Jan 30 '15 at 0:03




$begingroup$
@anorton Thank you for your help. I added some background information.
$endgroup$
– Audrey
Jan 30 '15 at 0:03












$begingroup$
@Audrey note that if one plane flies into west you can view that as a vector pointing to the left, that is in negative x-direction on a plane (-something,0). The plane that flies to the south, you can view as vector pointing down, or negative y-direction on a plane (0, -somethingelse). so you can apply pythagorean theorem something^2+somethingelse^2=result^2...
$endgroup$
– Loreno Heer
Jan 30 '15 at 0:22




$begingroup$
@Audrey note that if one plane flies into west you can view that as a vector pointing to the left, that is in negative x-direction on a plane (-something,0). The plane that flies to the south, you can view as vector pointing down, or negative y-direction on a plane (0, -somethingelse). so you can apply pythagorean theorem something^2+somethingelse^2=result^2...
$endgroup$
– Loreno Heer
Jan 30 '15 at 0:22










2 Answers
2






active

oldest

votes


















0












$begingroup$

Paths of the planes



$p_1(t) = (t cdot 300, 0)$



$p_2(t) = (0,(t-1/6) cdot (240))$



$s(t) = p_1(t) - p_2(t)$



$left.left(frac{d}{dt} |s(t)| right) rightvert_{t=.5} = ...$






share|cite|improve this answer











$endgroup$





















    0












    $begingroup$

    You have $x^2 + y^2 = z^2$
    your goal is to find $frac {dz}{dt}$



    $2zfrac {dz}{dt} = 2xfrac {dx}{dt} + 2yfrac {dy}{dt}$



    Now you will need to find the positions of the planes at the appointed time, to find $x,y,z$



    $x$ is the distance the west-bound airplane travels in 30 minutes.



    $y$ is the distance the south-bound airplane travels in 20 minutes.



    Use the Pythagorean theorem above to find $z.$



    distance = speed $times$ time



    You will need to convert units because the speed is in mph, and time is in minutes.



    And $frac {dx}{dt}, frac {dy}{dt}$ are the speeds of the planes.






    share|cite|improve this answer











    $endgroup$













      Your Answer





      StackExchange.ifUsing("editor", function () {
      return StackExchange.using("mathjaxEditing", function () {
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      });
      });
      }, "mathjax-editing");

      StackExchange.ready(function() {
      var channelOptions = {
      tags: "".split(" "),
      id: "69"
      };
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function() {
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled) {
      StackExchange.using("snippets", function() {
      createEditor();
      });
      }
      else {
      createEditor();
      }
      });

      function createEditor() {
      StackExchange.prepareEditor({
      heartbeatType: 'answer',
      autoActivateHeartbeat: false,
      convertImagesToLinks: true,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      imageUploader: {
      brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
      contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
      allowUrls: true
      },
      noCode: true, onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      });


      }
      });














      draft saved

      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1125762%2frelated-rates-problem-involving-airplanes%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      0












      $begingroup$

      Paths of the planes



      $p_1(t) = (t cdot 300, 0)$



      $p_2(t) = (0,(t-1/6) cdot (240))$



      $s(t) = p_1(t) - p_2(t)$



      $left.left(frac{d}{dt} |s(t)| right) rightvert_{t=.5} = ...$






      share|cite|improve this answer











      $endgroup$


















        0












        $begingroup$

        Paths of the planes



        $p_1(t) = (t cdot 300, 0)$



        $p_2(t) = (0,(t-1/6) cdot (240))$



        $s(t) = p_1(t) - p_2(t)$



        $left.left(frac{d}{dt} |s(t)| right) rightvert_{t=.5} = ...$






        share|cite|improve this answer











        $endgroup$
















          0












          0








          0





          $begingroup$

          Paths of the planes



          $p_1(t) = (t cdot 300, 0)$



          $p_2(t) = (0,(t-1/6) cdot (240))$



          $s(t) = p_1(t) - p_2(t)$



          $left.left(frac{d}{dt} |s(t)| right) rightvert_{t=.5} = ...$






          share|cite|improve this answer











          $endgroup$



          Paths of the planes



          $p_1(t) = (t cdot 300, 0)$



          $p_2(t) = (0,(t-1/6) cdot (240))$



          $s(t) = p_1(t) - p_2(t)$



          $left.left(frac{d}{dt} |s(t)| right) rightvert_{t=.5} = ...$







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 29 '15 at 23:43

























          answered Jan 29 '15 at 23:31









          Loreno HeerLoreno Heer

          3,34411534




          3,34411534























              0












              $begingroup$

              You have $x^2 + y^2 = z^2$
              your goal is to find $frac {dz}{dt}$



              $2zfrac {dz}{dt} = 2xfrac {dx}{dt} + 2yfrac {dy}{dt}$



              Now you will need to find the positions of the planes at the appointed time, to find $x,y,z$



              $x$ is the distance the west-bound airplane travels in 30 minutes.



              $y$ is the distance the south-bound airplane travels in 20 minutes.



              Use the Pythagorean theorem above to find $z.$



              distance = speed $times$ time



              You will need to convert units because the speed is in mph, and time is in minutes.



              And $frac {dx}{dt}, frac {dy}{dt}$ are the speeds of the planes.






              share|cite|improve this answer











              $endgroup$


















                0












                $begingroup$

                You have $x^2 + y^2 = z^2$
                your goal is to find $frac {dz}{dt}$



                $2zfrac {dz}{dt} = 2xfrac {dx}{dt} + 2yfrac {dy}{dt}$



                Now you will need to find the positions of the planes at the appointed time, to find $x,y,z$



                $x$ is the distance the west-bound airplane travels in 30 minutes.



                $y$ is the distance the south-bound airplane travels in 20 minutes.



                Use the Pythagorean theorem above to find $z.$



                distance = speed $times$ time



                You will need to convert units because the speed is in mph, and time is in minutes.



                And $frac {dx}{dt}, frac {dy}{dt}$ are the speeds of the planes.






                share|cite|improve this answer











                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  You have $x^2 + y^2 = z^2$
                  your goal is to find $frac {dz}{dt}$



                  $2zfrac {dz}{dt} = 2xfrac {dx}{dt} + 2yfrac {dy}{dt}$



                  Now you will need to find the positions of the planes at the appointed time, to find $x,y,z$



                  $x$ is the distance the west-bound airplane travels in 30 minutes.



                  $y$ is the distance the south-bound airplane travels in 20 minutes.



                  Use the Pythagorean theorem above to find $z.$



                  distance = speed $times$ time



                  You will need to convert units because the speed is in mph, and time is in minutes.



                  And $frac {dx}{dt}, frac {dy}{dt}$ are the speeds of the planes.






                  share|cite|improve this answer











                  $endgroup$



                  You have $x^2 + y^2 = z^2$
                  your goal is to find $frac {dz}{dt}$



                  $2zfrac {dz}{dt} = 2xfrac {dx}{dt} + 2yfrac {dy}{dt}$



                  Now you will need to find the positions of the planes at the appointed time, to find $x,y,z$



                  $x$ is the distance the west-bound airplane travels in 30 minutes.



                  $y$ is the distance the south-bound airplane travels in 20 minutes.



                  Use the Pythagorean theorem above to find $z.$



                  distance = speed $times$ time



                  You will need to convert units because the speed is in mph, and time is in minutes.



                  And $frac {dx}{dt}, frac {dy}{dt}$ are the speeds of the planes.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Feb 15 '18 at 22:48

























                  answered Feb 15 '18 at 22:42









                  Doug MDoug M

                  45.2k31954




                  45.2k31954






























                      draft saved

                      draft discarded




















































                      Thanks for contributing an answer to Mathematics Stack Exchange!


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid …



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      Use MathJax to format equations. MathJax reference.


                      To learn more, see our tips on writing great answers.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1125762%2frelated-rates-problem-involving-airplanes%23new-answer', 'question_page');
                      }
                      );

                      Post as a guest















                      Required, but never shown





















































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown

































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown







                      -

                      Popular posts from this blog

                      How to change which sound is reproduced for terminal bell?

                      Image
                      3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question
                      Read more

                      Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

                      Image
                      up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx
                      Read more

                      Can I use Tabulator js library in my java Spring + Thymeleaf project?

                      Image
                      0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"
                      Read more