You want to construct a box whose base length is 3 times the base width. The material used to build the top and bottom cost $10 per square foot and the material used to build the sides cost $6 per square foot. If the box must have a volume of 50 cubic feet determine the dimensions that will minimize the cost to build the box.
[[Click to start solving->First Step]]
<a data-flickr-embed="true" href="https://www.flickr.com/photos/specialtyboxes/7892904236/in/photolist-d2tcrN-d2tcNJ-d2tcKJ-8norS-dCqhj-os943R-XLH3kg-9zhtPP-YpxYYf-54fqNh-Mb7PaJ-MtDTBG-LmdT6y-dSJJMq-2hJi7UQ-J6pVx6-2gpVEms-2hFdy4W-c3eXQ-6GGUuQ-c3f75-zEg2SU-aWX1Xn-izDpH8-izDvuh-c3eDz-c3gKo-5D1JLL-izDEj1-Gapmh-qRpaME-29XbcUW-gzFt1-W5fr7y-Jj5Hi-GSLujT-fVK4W-2h1eFCX-2g9jJhP-NPbynt-AWSkV-miwJk3-47bjyc-L12ypi-ao3jGZ-29BZ9Xu-PZQWmo-FD9to2-QAfC6c-bgaLRz" title="Cardboard Box/RSC -SRA3"><img src="https://live.staticflickr.com/8172/7892904236_4303882bfb_q.jpg" width="150" height="150" alt="Cardboard Box/RSC -SRA3"></a><script async src="//embedr.flickr.com/assets/client-code.js" charset="utf-8"></script>What is your first step?
[[A. Write an equation that relates all of the information.->Write an equation that relates all of the information.]]
[[B. Write an equation for the variable being optimized.->Write an equation for the variable being optimized.]]
[[C. Write an equation for the variable being optimized and a secondary equation using the constraint.->Write an equation for the variable being optimized and a secondary equation using the constraint.]]
If having any trouble, review this video.
<iframe width="560" height="315" src="https://www.youtube.com/embed/HoIyp8Z11xM" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>Writing an equation that relates information found in the problem is a good start. When solving related rate problems this is the first step, however, when optimizing all of the information usually cannot be related into one equation in the fist step.
[[Click here to return and choose a different path->First Step]]Yes, you do need to write an equation for the variable being optimized, but that's not all. You also need to find a secondary equation relating to the contraint given in the problem.
[[Go Back->First Step]]Great work! When doing an optimization problem you usually need to write two equations, 1) an equation for what you are optimizing and 2) an equation related to the constraint given in the problem.
Which pair of equation make the most sense to use for this problem?
[[Optimization: Area = 6w^^2^^+6w^^2^^h+2wh
Constrains: 50 = lwh = 3w^^2^^h->Answer 1]]
[[Optimization: 50 = lwh = 3w^^2^^h
Constrains: Cost = 6w^^2^^(10)+6w^^2^^h(6)+2wh(6)->Answer 2]]
[[Optimization: Cost = 60w^^2^^+48wh
Constrains: 50 = lwh = 3w^^2^^h->Answer 3]]The equations you have chosen are a good start. Keep in mind what is being optimized and that you need to simplify. Is the variable or quantity that is being optimized represented in your equations.
[[Go back->Write an equation for the variable being optimized and a secondary equation using the constraint.]]The equations you have chosen are a good start. Keep in mind what is being optimized. What is the quantity that you are optimizing? Is it represented in your optimization equation?
[[Go back->Write an equation for the variable being optimized and a secondary equation using the constraint.]]Correct!
Now, you need to substitute your constraint equation into your optimization equation. This will result in a function of cost, C(w). Which of the fuctions below is your final cost fuction?
[[c(w) = 60w^^2^^+800w^^-1^^->Option 1]]
Or
[[c(w) = 60w^^2^^+200w->Option 2]]Great work finding the correct cost function. Your next step is to find c'(x). Which of the following is c'?
[[c'(x) = 120w-800w^^-2^^->Way 1]]
Or
[[c'(x) = 60w+200w^^-2^^->Way 2]]Your equation is close, however, is cost represented in your equation?
[[Click to go back->Answer 3]]Fantastic!
Now set c'(w) equal to zero and solve for w.
[[20/3->Path 1]]
or
[[1.8821->Path 2]]Be careful with your negative signs and don't forget to multiply.
[[Click to go back->Option 1]]Don't forget to take the cube root.
[[Click to go back->Way 1]]Great work! Now we need to interpret our results. What is your interpretation of w = 1.8821?
[[The least expensive box is made with dimentions 1.88x5.65x4.71.->Final Way 1]]
or
[[The largest box that can be made for $50 has dimentions 1.88x5.65x4.71.->Final Way 2]]Great work solving the problem.
<a data-flickr-embed="true" href="https://www.flickr.com/photos/erik_minnema/25587402401/in/photolist-EZ58hn-2bfVhVp-qAVwUF-22XEVZh-2h1P7S3-2gHsmjS-Ctt5qc-QoQats-7dgARG-MWWx3q-9ho1LA-H1UXd8-2dJwNnE-2hbhEqG-2hv9dPH-RLB73c-f5ax7i-Xs7HeD-MtW27j-21ZtQZq-BFNbVm-25iNBXV-FxRozZ-2dEbjcN-nCDTqF-QrD6zE-Cp1tbB-2bmJMZc-6mXPrH-wd12F1-7V7CDR-2aEEfxt-7tgXLY-SSNf8A-cxMaRY-ZCAWJv-FHR8HQ-BJc7SD-23s8nVV-4zGQpQ-2eYKUf-2eLt4J8-6juEDB-JPLHi2-oA6R4i-8YHt3g-7N7mr1-644Whv-oX28Fv-23ZJ14" title="Thumb up..."><img src="https://live.staticflickr.com/1601/25587402401_af3cc7b196_q.jpg" width="150" height="150" alt="Thumb up..."></a><script async src="//embedr.flickr.com/assets/client-code.js" charset="utf-8"></script>Be careful! Don't forget what you are trying to optimize?
[[Click to go back->Path 2]]