I still need help, thanks.
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 ?
- what is the volume of the prism you need, with height and side ?
- what is the lateral area of this prism?
- express the price in terms of and .
- given the volume of the prism, express in terms of (and , fixed). Use this to express the price in terms of (and of , which is fixed).
- study the price function you have obtained (depending on ): for which is it minimum?, what is this minimum value? What is the value of corresponding to that ?
It is , you know that.
Volume of a right prism = (area of one end) times (height)- what is the volume of the prism you need, with height and side ?
In the present case, , i.e. .
The sides of the prism are rectangles with length and width and . So the area of each side is . The area of the floor is and the area of the ceiling is .- what is the lateral area of this prism?
Hence .- express the price in terms of and .
Can you go on? Express in terms of using the equation about the volume, replace in the price function, and then differentiate with respect to to study the variations.
If you're happy with this forum, you should rather consider donating, this could benefit to every member.
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.
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 , so that . Replace in , use , and you get the equation I wrote.
Differentiate this equation: . 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 . This gives you the value I gave.
Now, from this value for , you find the price and the length , 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.