# Thread: Production Cost Calculus Problem???

I've been trying to figure out this problem for a while now and I'm not understanding how to do it and can't get the correct answers, can someone explain it to me please? Thanks...

Question Details:
I need to build a tent to be in the shape of a right prism whose ends are equilateral triangles, with the door to the tent on one of the triangles. I need the volume to be 2.2 cubic meters. The material to be used for the flooring costs $14 per square meter, but the material making up the ends and the top of the tent is only$10 per square meter.

What is minimum cost for making this tent? What should the dimensions be to have a minimum production cost? What will the minimum cost be? How much material should I order for each tent?

2. I still need help, thanks.

3. Originally Posted by yoyo
I've been trying to figure out this problem for a while now and I'm not understanding how to do it and can't get the correct answers, can someone explain it to me please? Thanks...

Question Details:
I need to build a tent to be in the shape of a right prism whose ends are equilateral triangles, with the door to the tent on one of the triangles. I need the volume to be 2.2 cubic meters. The material to be used for the flooring costs $14 per square meter, but the material making up the ends and the top of the tent is only$10 per square meter.

What is minimum cost for making this tent? What should the dimensions be to have a minimum production cost? What will the minimum cost be? How much material should I order for each tent?
Split the problem into parts:
- what is the area of an equilateral triangle with side $a$?
- what is the volume of the prism you need, with height $h$ and side $a$?
- what is the lateral area of this prism?
- express the price $P$ in terms of $a$ and $h$.
- given the volume $V$ of the prism, express $h$ in terms of $a$ (and $V$, fixed). Use this to express the price in terms of $a$ (and of $V$, which is fixed).
- study the price function you have obtained (depending on $a\geq 0$): for which $a$ is it minimum?, what is this minimum value? What is the value of $h$ corresponding to that $a$?

4. Originally Posted by Laurent
Split the problem into parts:
- what is the area of an equilateral triangle with side $a$?
- what is the volume of the prism you need, with height $h$ and side $a$?
- what is the lateral area of this prism?
- express the price $P$ in terms of $a$ and $h$.
- given the volume $V$ of the prism, express $h$ in terms of $a$ (and $V$, fixed). Use this to express the price in terms of $a$ (and of $V$, which is fixed).
- study the price function you have obtained (depending on $a\geq 0$): for which $a$ is it minimum?, what is this minimum value? What is the value of $h$ corresponding to that $a$?

Thanks, I'll give it a try.

5. Can someone show me step by step with equations please. I'm willing to give a $15.00 iTunes gift card to anyone that can show me exactly how to do it with the equations and show me the correct answers. Thanks. 6. Originally Posted by Laurent Split the problem into parts: - what is the area of an equilateral triangle with side $a$? It is $a^2\frac{\sqrt{3}}{4}$, you know that. - what is the volume of the prism you need, with height $h$ and side $a$? Volume of a right prism = (area of one end) times (height) In the present case, $V=a^2h\frac{\sqrt{3}}{4}$, i.e. $2.2=a^2h\frac{\sqrt{3}}{4}$. - what is the lateral area of this prism? The sides of the prism are rectangles with length and width $a$ and $h$. So the area of each side is $ah$. The area of the floor is $ah$ and the area of the ceiling is $2ah$. - express the price $P$ in terms of $a$ and $h$. Hence $P=14\mbox{(area of the floor)}+10\mbox{(area of top and ends)}=14 ah + 10\left(2ah+a^2\frac{\sqrt{3}}{4}\right)$. Can you go on? Express $h$ in terms of $a$ using the equation about the volume, replace in the price function, and then differentiate $P$ with respect to $a$ to study the variations. Originally Posted by yoyo Can someone show me step by step with equations please. I'm willing to give a$15.00 iTunes gift card to anyone that can show me exactly how to do it with the equations and show me the correct answers. Thanks.
If you're happy with this forum, you should rather consider donating, this could benefit to every member.

7. I think you should find $P\simeq\frac{172.7}{a}+4.33 a^2$, hence $P'(a)=0$ when $a^3\simeq 19.9$, hence $a=2.71{\rm m}$, from which we deduce $h=0.69{\rm m}$, and the price for these dimensions is $P\simeq \95.5$. (Up to possible computation errors)

8. Originally Posted by Laurent
I think you should find $P\simeq\frac{172.7}{a}+4.33 a^2$, hence $P'(a)=0$ when $a^3\simeq 19.9$, hence $a=2.71{\rm m}$, from which we deduce $h=0.69{\rm m}$, and the price for these dimensions is $P\simeq \95.5$. (Up to possible computation errors)
Hummm... I'm lost, So your saying that the dimesions are h=0.69m and a=2.71m???? and the minimum cost to make the tent is \$95.5???? How did you get these, (math wise)---can you show your calculations?. How much material should be order for each tent??? Also you could go ahead and show me how to express h in terms of a using the equation about the volume, replace in the price function, and then differentiate P with respect to a to study the variations

Thanks for your quick response, I'll donate some money to this site if you've rather me too instead of giving you the gift card.

9. Originally Posted by yoyo
How did you get these, (math wise)---can you show your calculations?. How much material should be order for each tent??? Also you could go ahead and show me how to express h in terms of a using the equation about the volume, replace in the price function, and then differentiate P with respect to a to study the variations
Did you give it try? I was giving my answers so you can compare with yours.

There's almost nothing more to do: we know $V=a^2h\frac{\sqrt{3}}{4}$, so that $h=\frac{4V}{a^2\sqrt{3}}$. Replace in $P$, use $V=2.2$, and you get the equation I wrote.

Differentiate this equation: $P'(a)=-\frac{172.7}{a^2}+8.66 a = \frac{8.66a^3-172.7}{a^3}$. Study the sign of this derivative, it gives you the variations of the function (where it is increasing or decreasing). In particular, you can find that the price is minimum when $P'(a)=0$. This gives you the value I gave.

Now, from this value for $a$, you find the price $P$ and the length $h$, and the area of material required, by replacing in the appropriate equations. I'm sure you can do that, and it won't help you if I do it myself.

### production cost problem pre

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