x and y are similar shapes. the total surface area of x is 900cm squared. the total surface area of y is 1600cm squared. the volume of x is 540cm squared. calculate the volume of y
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(I'm going to assume that you meant the volume of x to be 540cm.3 )
Let's assume that we could take all the "surfaces" of x and lay them flat on a table.
Note that, if x's surface area is 900cm2, and it volume is 540cm3, the other dimension must be .6cm.
And the "scaling" factor between the larger object y and the smaller similar object x is given by SQRT(y's surface area/ x's surface area) = SQRT(1600/900) = 40/30 = 4/3
To see that this is true, let's suppose that the surface area of x is just a square with a side = 30cm - it may not be a square, but let's suppose that it is !! Again, let's suppose that the surface area of y is a square, too, with a side of 40cm. Then, the scaling factor is 4/3....for each dimension of x, the same dimension in y is 4/3 as long......since they're "similar" objects.
Well, if that's true, the other dimension of our hypothetical y must be = (4/3) * .6cm = .8cm.
So , if we could take y's "surfaces" and lay them flat, too, it's total volume would be !600cm * .8cm. = 1280cm3
Finally, no matter what the dimensions of x, each dimension of y is (4/3) as much. So, if the width (w) * length (l) * height (h) of x = volume = 540cm3, then the dimensions of y = (w*4/3) (l*4/3) (h*4/3) = (w* l * h) * (4/3)3 =
(volume of x) * (scaling factor)3 = (540cm3) * (4/3)3 = 1280cm3.......which is just what we thought!!!
, and the impossibility of this undertaking follows from the fact that pi is a transcendental (non-algebraic and therefore non-constructible) number. If the problem of the quadrature of the circle is solved using only compass and straightedge, then an algebraic value of pi would be found, which is impossible. Johann Heinrich Lambert conjectured that pi was transcendental in 1768 in the same paper in which he proved its irrationality, even before the existence of transcendental numbers was proven. It was not until 1882 that Ferdinand von Lindemann proved its transcendence.
