I’m just trynna get points
The dimensions of a box that have the minium surface area for a given Volume is such that it is a cube. This is the three dimensions are equal:
V = x*y*z , x=y=z => V = x^3, that will let you solve for x,
x = ∛(V) = ∛(250cm^3) = 6.30 cm.
Answer: 6.30 cm * 6.30cm * 6.30cm. This is a cube of side 6.30cm.
The demonstration of that the shape the minimize the volume of a box is cubic (all the dimensions equal) corresponds to a higher level (multivariable calculus).
I guess it is not the intention of the problem that you prove or even know how to prove it (unless you are taking an advanced course).
Nevertheless, the way to do it is starting by stating the equations for surface and apply two variable derivation to optimize (minimize) the surface.
You do not need to follow with next part if you do not need to understand how to show that the cube is the shape that minimize the surface.
If you call x, y, z the three dimensions, the surface is:
S = 2xy + 2xz + 2yz (two faces xy, two faces xz and two faces yz).
Now use the Volumen formula to eliminate one variable, let's say z:
V = x*y*z => z = V /(x*y)
=> S = 2xy + 2x [V/(xy)[ + 2y[V/(xy)] = 2xy + 2V/y + 2V/x
Now find dS, which needs the use of partial derivatives. It drives to:
dS = [2y - 2V/(x^2)] dx + [2x - 2V/(y^2) ] dy = 0
By the properties of the total diferentiation you have that:
2y - 2V/(x^2) = 0 and 2x - 2V/(y^2) = 0
2y - 2V/(x^2) = 0 => V = y*x^2
2x - 2V/(y^2) = 0 => V = x*y^2
=> y*x^2 = x*y^2 => y*x^2 - x*y^2 = xy (x - y) = 0 => x = y
Now that you have shown that x = y.
You can rewrite the equation for S and derive it again:
S = 2xy + 2V/y + 2V/x, x = y => S = 2x^2 + 2V/x + 2V/x = 2x^2 + 4V/x
Now find S'
S' = 4x - 4V/(x^2) = 0 => V/(x^2) = x => V =x^3.
Which is the proof that the box is cubic.
It will take 21319.2 hrs.
4.23 x60=253.8
253.8 x 84= 21319.2
hope this helps
Hello! Sauce B and C do not have he highest numbers and they were not any of the highest rated sauces. Therefore, A and B are eliminated. Sauce D was rated 4 in terms of cost, making it the best price and Sauce C has a score of 1, making it the worst price. Sauce D would be good to buy it cost was a big concern, but it was rated 2nd in the family anyways, so it would be fine for them. The answer is C.
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