![]() Is equal to 180 over 40, which is the same Of this equation by x squared and we would get 40x to And I could write itĪs 180 over x squared. To the negative 2 equal 0? Well, we could add the 180x to This derivative is equal to 0 in our search for And actually, if xĮquals 0 then our height is undefined as well. We would have no volume atĪll, so it would not work out. So we don't want to worryĪbout that critical point. Not interesting to us as a critical pointīecause then we're going to have a degenerate. It's defined for all x except for x equaling 0. Times x minus 180 times x to the negative 2 power. Our cost with respect to x is going to be equal to 40 Is undefined or equal to 0, and those are ourĬandidate critical points. We take the derivative, figure out where the derivative To optimize, we justĬritical points are a minimum or a maximum value. So we finally have costĪs a function of x. So it's going to be plusġ80 times, let's see, x times x to the negativeĢ, 180x to the negative x to the negative 1 power. Times 5 is 150 plus another 30 is going to be 180. Of x is going to be 20x squared 36 times 5. Going to be equal to 20 times x squared plus 36 times And then we can substituteīack right over here. Or we could say that h isĮqual to 5 over x squared. Or another way of sayingĢx squared times h needs to be equal to 10. Times the length times 2x times the height times h So how can we do that? How can we express hĪs a function of x? Well, we know that the volume But if we want to optimizeĪs a function of x. With respect to one variable, and maybe I'll say let's To be 2 times 6, which is 12 times 2 is 24xh plus 24xh. This is right over here, this is the cost of the sides. Of the panels is going to be $6 per square Panel right over here and we have this side Would be the cost of one of these side panels. Over here and this side over here, which haveĪrea of x times h. Sides are going to have different dimensions. And now what's going toīe the cost of the sides? Well, the different Well, what's theĪrea of the base? Well, it's going to be the ![]() Is going to be $10 times- I'll just write 10. This container going to be? So the cost is going to beĮqual to the cost of the base. But x only gives us theĭimension for height. So let's see if we canĬome up with a value or how much this box would cost ![]() So the material over hereĬosts $6 per meter squared. Material for the sides costs $6 per square meter. Here, that material costs $10 per square meter. Over here- if I was transparent I couldĬontinue to draw it down here. Material for the base costs $10 per square meter. The volume needs to beĮqual to 10 meters cubed. And so what do they tell us? They tell us that the volume And then I could draw-Īnd since it's open top, I can see through, I could see Open storage container, this open rectangular Find the cost of the materialįor the cheapest container. Material for the sidesĬosts $6 per square meter. Material for the baseĬosts $10 per square meter. The $4 per foot side Y=340/√(130)=29.820 feetĬontainer with an open top needs to have a volume Now to find C plug X into the cost equation. In either case we don't have to do that in this problem, so moving on. You could either do the second derivative test to find out whether they are maximums or minimums, or you could simply plug into the cost equation and find out which one is cheaper. If there were multiple positive X's it would be simple to test. Right away you can discard the negative answer because X is a length and you cannot have negative length, so X=2√(130). Next, set dc/dx equal to 0 this will allow you to find the x value for all mins and maxes in the original equation. Simply isolate the Y variable on one side of the first equation so you can substitute the expression it's equal to into the second equation.Ĭhange the 6240/x to 6240X^-1 to make deriving easier. To move on you need a single equation with only 2 variables. You can then assign the $6 cost to one variable and the $4 cost to the other. Perimeter is equal to twice the length + twice the width. the cost of the materials is going to be determined by the perimeter of the space. ![]() You also know that c is the combined cost of the materials. X=width of the space, Y=length of the space, and C=cost of materials.īecause you know that the area is 780 square feet, you know that 780 is the product of x and y.
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