This summer Laura hopes to raise rabbits for "sale". The rabbit hutch is in the shape of a equilateral triangle-ended prism with a total volume of...
3m^3. If the cost of the wood for the side walls is $10/m^2 and the wood for the floor is $20/m^2. Determine the dimensions of the hutch that keeps the total cost of the wood to a minimum.
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Notwithstanding the first answer given, I believe that there is no way to change the size of the floor relative to the sizes of the walls. This is because the triangles at either end of the prism are equilateral, such that each wall and the floor are all identical rectangles.
For a triangular prism, the volume is equal to the area of one triangle times the height (distance between the two triangular ends) = ½(abh).
The surface area (and quantity of wood needed to construct) = ab +(s1 + s2 + s3)b.
Since the triangular ends are equilateral, s1 = s2 = s3.
Therefore the surface area now becomes ab + 3b.
Since the triangle is equilateral, the structure will consist of 3 identical rectangular pieces of wood, two of which are walls tilted against each other, and one floor piece.
For any given solid, the ratio of surface area to volume is constant, regardless of the overall dimensions.
The floor wood area will comprise 1/3 of the total wood area. If total wood area is X, wood cost of floor will be 20X/3. Cost of wall wood is 2(10X/3) = 20X/3. Total wood cost = 40X/3.
Changing the dimensions of the hutch will not alter the fact that the floor wood will always cost the same as the combined cost of the wall pieces. The floor wood is twice as expensive as the wall pieces, so for the same cost, you can have one floor piece or two wall pieces.
The identical size of floor piece area to sidewall area is constant, and not alterable by changing dimensions of the hutch.
By this reasoning, I believe that there is no solution to the problem.
Remember what is the volume of a triangular prism.
V = Area of the base*height
Area of the base is the area of an equilateral triangle. Use the next notation for the side of triangle: side = x.
Area of triangle = `(x*x*sin 60)/2 = (x^2*sqrt 3)/4`
`V = (x^2*sqrt 3)*h/4`
`3 = (x^2*sqrt 3)*h/4 =gt 12 = (x^2*sqrt 3)*h =gt h = 12/x^2*sqrt3 =gt h = 12sqrt3/3x^2 =gt h = 4sqrt3/x^2`
Since the cost of wood for the side walls is 10$/m => the cost of the side walls surfaces is: `30x*h $/m`
Since the cost of wood for the floors is 20$/m => the cost of the floors surfaces is: `2*20*(x^2*sqrt 3)/4 $/m = 10sqrt3*x^2$/m`
`` Write the equation of the total cost of the rabbit hutch.
`C(x) = 30x*h + 10sqrt3*x^2`
Substitute h by `4sqrt3/x^2.`
`C(x) = 30x*4sqrt3/x^2 + 10sqrt3*x^2`
`C(x) = 120sqrt3/x + 10sqrt3*x^2`
For the cost to be kept at minimum value, you should find the derivative of the cost function and to cancel it out.
`C'(x) = -120sqrt3/x^2 + 20 sqrt3 x`
Put `C'(x) = 0 =gt -120sqrt3/x^2 + 20 sqrt3 x = 0`
`-120sqrt3 +20 sqrt3 x^3 = 0`
`20sqrt3(-6 + x^3) = 0 =gt x^3-6 = 0 =gt (x-root(3)(6))(x^2 + x*root(3)(6) + root(3)(36)) = 0`
`` `x = root(3)(6)`meters
`h = (4 root(3)(3))/x^2 = (4 root(3)(18))/6 = (2root(3)(18))/3 ` meters
The dimensions that keep the cost at minimum are x = `root(3)(6)` meters and h = `(2 root(3)(18))/3` meters.
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