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The vitamins in a spherical fruit are concentrated in the outer skin. How much more of...

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The vitamins in a spherical fruit are concentrated in the outer skin. How much more of vitamins does a person get if instead of buying one 100 g fruit 4 25 g fruits are bought.

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The vitamins in the fruit are concentrated in the outer skin and the fruit is spherical in shape. Assume that the density of the fruit is the same irrespective of the size.

If the volume of a 100 g fruit is V, the volume of a 25 g fruit is V/4. The volume of a sphere with radius r is `(4/3)*pi*r^3` and the surface area of the sphere is `4*pi*r^2` .

Let `(4/3)*pi*r^3 = V`

=> `r = root(3)((3/4)*V/pi)`

The surface area is `4*pi*(root(3)((3/4)*V/pi))^2`

The surface area of 4 25 g fruits is `4*4*pi*(root(3)((3/4)*V/(4*pi)))^2`

Dividing the area of the 4 25 g fruits by that of the 100 g fruit gives:

`(4*4*pi*(root(3)((3/4)*V/(4*pi)))^2)/(4*pi*(root(3)((3/4)*V/pi))^2)`

= `4*(root(3)(1/(4)))^2`

= `4*1/4^(2/3)`

`~~ 1.587`

The person gets approximately 58.7% more vitamins if 4 smaller fruit are bought.

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