A sculptor works with large wooden cuboids (rectangular prisms), cuts them into three smaller cuboids and then paints them black, white and grey. The volume of the black cuboid is always twice the volume of the white one and the volume of the white cuboid is always twice the volume of the grey one.
The attached diagram shows how this works. The cuts are always parallel to the face marked with w and h. We will always denote the surface areas of the separated cuboids by A1(black ) A2 (white) A3 (grey)
a)the sculptor is given a block for which l=1, w=6, h=3 Show that with these values the ratio of A1 to A2 is 3:2
b) show that for any cuboid the sculptor cuts A1 is always less than 2(A2)
c)the sculptor wants a cuboid with h=w which will give A1=3(A3). what would be the value of l in terms of w
d) the sclptor decides she would like to have A1:A3 = 4:3. If this is to be the case find a formula linking l,w,h and give an example of the dimension of a cuboid before it is cut for which this will work.
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The surface area of a cuboid with dimensions W, D, H is:
Originally Posted by shosho
A(W,D,H) = 2W*D + 2W*H + 2D*H
That is there are six faces two of dimensions W,D, 2 of dimensions W,H
and 2 of dimensions D, H.
Now the surface areas of the three cuboids are:
A3 = 2w*l + 2w*h + 2l*h
A2 = 2w*(2*l) + 2w*h + 2(2*l)*h
A1 = 2w*(4*l) + 2w*h + 2(4*l)*h