Having difficulty understanding how to find the maximum volume with cost constraint.












0












$begingroup$


We have 1000USD to buy the materials to build a box whose base length is seven times the base width and has no top. If the material for the sides costs 10USD/cm2 and the material of the bottom costs 15USD/cm2 determine the dimensions of the box that will maximise the enclosed volume.



Question from: http://tutorial.math.lamar.edu/ProblemsNS/CalcI/Optimization.aspx



I've tried solving it but each time I get a different answer and it doesn't make any sense. I got 164.65cm3 as an answer...






calculus optimization volume







share|cite|improve this question











$endgroup$












  • $begingroup$
    Call base length $L$, weight $w$ and height $h$. Now set up equations based on what you know and an expression for what you want to maximize.
    $endgroup$
    – mathreadler
    Jan 7 at 14:49










  • $begingroup$
    math.meta.stackexchange.com/questions/9959/…
    $endgroup$
    – John Douma
    Jan 7 at 14:56










  • $begingroup$
    @mathreadler i did that and I'm pretty sure my following out is right but my answer never makes sense.
    $endgroup$
    – Struggling
    Jan 7 at 15:05










  • $begingroup$
    Show us the work you have done, we can see where you went wrong.
    $endgroup$
    – Calvin Godfrey
    Jan 7 at 15:08










  • $begingroup$
    Show us your calculations and we'll critique them.
    $endgroup$
    – saulspatz
    Jan 7 at 15:08
















0












$begingroup$


We have 1000USD to buy the materials to build a box whose base length is seven times the base width and has no top. If the material for the sides costs 10USD/cm2 and the material of the bottom costs 15USD/cm2 determine the dimensions of the box that will maximise the enclosed volume.



Question from: http://tutorial.math.lamar.edu/ProblemsNS/CalcI/Optimization.aspx



I've tried solving it but each time I get a different answer and it doesn't make any sense. I got 164.65cm3 as an answer...






calculus optimization volume







share|cite|improve this question











$endgroup$












  • $begingroup$
    Call base length $L$, weight $w$ and height $h$. Now set up equations based on what you know and an expression for what you want to maximize.
    $endgroup$
    – mathreadler
    Jan 7 at 14:49










  • $begingroup$
    math.meta.stackexchange.com/questions/9959/…
    $endgroup$
    – John Douma
    Jan 7 at 14:56










  • $begingroup$
    @mathreadler i did that and I'm pretty sure my following out is right but my answer never makes sense.
    $endgroup$
    – Struggling
    Jan 7 at 15:05










  • $begingroup$
    Show us the work you have done, we can see where you went wrong.
    $endgroup$
    – Calvin Godfrey
    Jan 7 at 15:08










  • $begingroup$
    Show us your calculations and we'll critique them.
    $endgroup$
    – saulspatz
    Jan 7 at 15:08














0












0








0





$begingroup$


We have 1000USD to buy the materials to build a box whose base length is seven times the base width and has no top. If the material for the sides costs 10USD/cm2 and the material of the bottom costs 15USD/cm2 determine the dimensions of the box that will maximise the enclosed volume.



Question from: http://tutorial.math.lamar.edu/ProblemsNS/CalcI/Optimization.aspx



I've tried solving it but each time I get a different answer and it doesn't make any sense. I got 164.65cm3 as an answer...






calculus optimization volume







share|cite|improve this question











$endgroup$




We have 1000USD to buy the materials to build a box whose base length is seven times the base width and has no top. If the material for the sides costs 10USD/cm2 and the material of the bottom costs 15USD/cm2 determine the dimensions of the box that will maximise the enclosed volume.



Question from: http://tutorial.math.lamar.edu/ProblemsNS/CalcI/Optimization.aspx



I've tried solving it but each time I get a different answer and it doesn't make any sense. I got 164.65cm3 as an answer...






calculus optimization volume




calculus optimization volume






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 7 at 15:45







Struggling

















asked Jan 7 at 14:39









StrugglingStruggling

42




42












  • $begingroup$
    Call base length $L$, weight $w$ and height $h$. Now set up equations based on what you know and an expression for what you want to maximize.
    $endgroup$
    – mathreadler
    Jan 7 at 14:49










  • $begingroup$
    math.meta.stackexchange.com/questions/9959/…
    $endgroup$
    – John Douma
    Jan 7 at 14:56










  • $begingroup$
    @mathreadler i did that and I'm pretty sure my following out is right but my answer never makes sense.
    $endgroup$
    – Struggling
    Jan 7 at 15:05










  • $begingroup$
    Show us the work you have done, we can see where you went wrong.
    $endgroup$
    – Calvin Godfrey
    Jan 7 at 15:08










  • $begingroup$
    Show us your calculations and we'll critique them.
    $endgroup$
    – saulspatz
    Jan 7 at 15:08


















  • $begingroup$
    Call base length $L$, weight $w$ and height $h$. Now set up equations based on what you know and an expression for what you want to maximize.
    $endgroup$
    – mathreadler
    Jan 7 at 14:49










  • $begingroup$
    math.meta.stackexchange.com/questions/9959/…
    $endgroup$
    – John Douma
    Jan 7 at 14:56










  • $begingroup$
    @mathreadler i did that and I'm pretty sure my following out is right but my answer never makes sense.
    $endgroup$
    – Struggling
    Jan 7 at 15:05










  • $begingroup$
    Show us the work you have done, we can see where you went wrong.
    $endgroup$
    – Calvin Godfrey
    Jan 7 at 15:08










  • $begingroup$
    Show us your calculations and we'll critique them.
    $endgroup$
    – saulspatz
    Jan 7 at 15:08
















$begingroup$
Call base length $L$, weight $w$ and height $h$. Now set up equations based on what you know and an expression for what you want to maximize.
$endgroup$
– mathreadler
Jan 7 at 14:49




$begingroup$
Call base length $L$, weight $w$ and height $h$. Now set up equations based on what you know and an expression for what you want to maximize.
$endgroup$
– mathreadler
Jan 7 at 14:49












$begingroup$
math.meta.stackexchange.com/questions/9959/…
$endgroup$
– John Douma
Jan 7 at 14:56




$begingroup$
math.meta.stackexchange.com/questions/9959/…
$endgroup$
– John Douma
Jan 7 at 14:56












$begingroup$
@mathreadler i did that and I'm pretty sure my following out is right but my answer never makes sense.
$endgroup$
– Struggling
Jan 7 at 15:05




$begingroup$
@mathreadler i did that and I'm pretty sure my following out is right but my answer never makes sense.
$endgroup$
– Struggling
Jan 7 at 15:05












$begingroup$
Show us the work you have done, we can see where you went wrong.
$endgroup$
– Calvin Godfrey
Jan 7 at 15:08




$begingroup$
Show us the work you have done, we can see where you went wrong.
$endgroup$
– Calvin Godfrey
Jan 7 at 15:08












$begingroup$
Show us your calculations and we'll critique them.
$endgroup$
– saulspatz
Jan 7 at 15:08




$begingroup$
Show us your calculations and we'll critique them.
$endgroup$
– saulspatz
Jan 7 at 15:08










1 Answer
1






active

oldest

votes


















0












$begingroup$

This is more a hint than a full solution! I can only help further if you can show what you are actually stuck on.



You can setup the problem as mathreadler suggested. Let $L$ be the length of the box, $H$ be the height of the box and $W$ the width of the box. Furthermore we define $c_b$ to be the cost of the bottom per unit area and $c_s$ the cost of the side per unit area.



Now we notice that $L = 7W$. We can now setup equations for the cost and the volume of the box.



The total volume, V, is given by:
$$ V = L times W times H = 7W^2H $$



The total cost, C, is given by (where A denotes area):
$$ C = A_{bottom}c_b + A_{side}c_s = 7W^2c_b + (W+7W+W+7W)Hc_s = 7W^2c_b + 16WHc_s$$



The total cost is constrained:
$$ C = 7W^2c_b + 16WHc_s = 105W^2 + 160WH leq 1000 $$



You now have to balance the width and hight of the box in such a way that the constraint is met, and the volume is as large as possible. You can do this using derivatives. Hopefully you can now do some sensible calculations






share|cite|improve this answer









$endgroup$













  • $begingroup$
    could you please go through my answer and see where I went wrong? I don't see what I did wrong but the answer doesn't make sense to me... docs.google.com/document/d/…
    $endgroup$
    – Struggling
    Jan 7 at 16:33














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1 Answer
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active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









0












$begingroup$

This is more a hint than a full solution! I can only help further if you can show what you are actually stuck on.



You can setup the problem as mathreadler suggested. Let $L$ be the length of the box, $H$ be the height of the box and $W$ the width of the box. Furthermore we define $c_b$ to be the cost of the bottom per unit area and $c_s$ the cost of the side per unit area.



Now we notice that $L = 7W$. We can now setup equations for the cost and the volume of the box.



The total volume, V, is given by:
$$ V = L times W times H = 7W^2H $$



The total cost, C, is given by (where A denotes area):
$$ C = A_{bottom}c_b + A_{side}c_s = 7W^2c_b + (W+7W+W+7W)Hc_s = 7W^2c_b + 16WHc_s$$



The total cost is constrained:
$$ C = 7W^2c_b + 16WHc_s = 105W^2 + 160WH leq 1000 $$



You now have to balance the width and hight of the box in such a way that the constraint is met, and the volume is as large as possible. You can do this using derivatives. Hopefully you can now do some sensible calculations






share|cite|improve this answer









$endgroup$













  • $begingroup$
    could you please go through my answer and see where I went wrong? I don't see what I did wrong but the answer doesn't make sense to me... docs.google.com/document/d/…
    $endgroup$
    – Struggling
    Jan 7 at 16:33


















0












$begingroup$

This is more a hint than a full solution! I can only help further if you can show what you are actually stuck on.



You can setup the problem as mathreadler suggested. Let $L$ be the length of the box, $H$ be the height of the box and $W$ the width of the box. Furthermore we define $c_b$ to be the cost of the bottom per unit area and $c_s$ the cost of the side per unit area.



Now we notice that $L = 7W$. We can now setup equations for the cost and the volume of the box.



The total volume, V, is given by:
$$ V = L times W times H = 7W^2H $$



The total cost, C, is given by (where A denotes area):
$$ C = A_{bottom}c_b + A_{side}c_s = 7W^2c_b + (W+7W+W+7W)Hc_s = 7W^2c_b + 16WHc_s$$



The total cost is constrained:
$$ C = 7W^2c_b + 16WHc_s = 105W^2 + 160WH leq 1000 $$



You now have to balance the width and hight of the box in such a way that the constraint is met, and the volume is as large as possible. You can do this using derivatives. Hopefully you can now do some sensible calculations






share|cite|improve this answer









$endgroup$













  • $begingroup$
    could you please go through my answer and see where I went wrong? I don't see what I did wrong but the answer doesn't make sense to me... docs.google.com/document/d/…
    $endgroup$
    – Struggling
    Jan 7 at 16:33
















0












0








0





$begingroup$

This is more a hint than a full solution! I can only help further if you can show what you are actually stuck on.



You can setup the problem as mathreadler suggested. Let $L$ be the length of the box, $H$ be the height of the box and $W$ the width of the box. Furthermore we define $c_b$ to be the cost of the bottom per unit area and $c_s$ the cost of the side per unit area.



Now we notice that $L = 7W$. We can now setup equations for the cost and the volume of the box.



The total volume, V, is given by:
$$ V = L times W times H = 7W^2H $$



The total cost, C, is given by (where A denotes area):
$$ C = A_{bottom}c_b + A_{side}c_s = 7W^2c_b + (W+7W+W+7W)Hc_s = 7W^2c_b + 16WHc_s$$



The total cost is constrained:
$$ C = 7W^2c_b + 16WHc_s = 105W^2 + 160WH leq 1000 $$



You now have to balance the width and hight of the box in such a way that the constraint is met, and the volume is as large as possible. You can do this using derivatives. Hopefully you can now do some sensible calculations






share|cite|improve this answer









$endgroup$



This is more a hint than a full solution! I can only help further if you can show what you are actually stuck on.



You can setup the problem as mathreadler suggested. Let $L$ be the length of the box, $H$ be the height of the box and $W$ the width of the box. Furthermore we define $c_b$ to be the cost of the bottom per unit area and $c_s$ the cost of the side per unit area.



Now we notice that $L = 7W$. We can now setup equations for the cost and the volume of the box.



The total volume, V, is given by:
$$ V = L times W times H = 7W^2H $$



The total cost, C, is given by (where A denotes area):
$$ C = A_{bottom}c_b + A_{side}c_s = 7W^2c_b + (W+7W+W+7W)Hc_s = 7W^2c_b + 16WHc_s$$



The total cost is constrained:
$$ C = 7W^2c_b + 16WHc_s = 105W^2 + 160WH leq 1000 $$



You now have to balance the width and hight of the box in such a way that the constraint is met, and the volume is as large as possible. You can do this using derivatives. Hopefully you can now do some sensible calculations







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 7 at 15:28









Tim DiklandTim Dikland

38319




38319












  • $begingroup$
    could you please go through my answer and see where I went wrong? I don't see what I did wrong but the answer doesn't make sense to me... docs.google.com/document/d/…
    $endgroup$
    – Struggling
    Jan 7 at 16:33




















  • $begingroup$
    could you please go through my answer and see where I went wrong? I don't see what I did wrong but the answer doesn't make sense to me... docs.google.com/document/d/…
    $endgroup$
    – Struggling
    Jan 7 at 16:33


















$begingroup$
could you please go through my answer and see where I went wrong? I don't see what I did wrong but the answer doesn't make sense to me... docs.google.com/document/d/…
$endgroup$
– Struggling
Jan 7 at 16:33






$begingroup$
could you please go through my answer and see where I went wrong? I don't see what I did wrong but the answer doesn't make sense to me... docs.google.com/document/d/…
$endgroup$
– Struggling
Jan 7 at 16:33




















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