The lengths of edges of rectangular box are x,y,z?x,y,z are diff. functions and s is diagonal of box. How is related ds/dt to dx/dt,dy/dt,dz,dt?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the length of diagonal of the box, hence, you need to find the diagonal of rectangular base first and then you need to find the diagonal of rectangular box.

Supposing that x and y are the length and the width of rectangular base, then diagonal is the square root of sum of squares of width and length such that:

`d = sqrt(x^2 + y^2)`

You need to find diagonal of rectangular box, thus you need to use Pythagorean theorem in the right triangle that has as lengths of legs z and d.

`s = sqrt(d^2 + z^2)`

Substituting `x^2 + y^2`  for d yields:

`s = sqrt(x^2 + y^2 + z^2)`

You need to raise to square both sides such that:

`s^2 = x^2 + y^2 + z^2`

You need to differentiate the function s with respect to t such that:

`2s(ds)/(dt) = 2x(dx)/(dt) + 2y(dy)/(dt) + 2z(dz)/(dt)`

You need to divide by 2s both sides such that:

`(ds)/(dt) = (x/s)(dx)/(dt) + (y/s)(dy)/(dt) + (z/s)(dz)/(dt)` 

You need to substitute `sqrt(x^2 + y^2 + z^2)`  for s such that:

`(ds)/(dt) = (x/(sqrt(x^2 + y^2 + z^2)))(dx)/(dt) + (y/(sqrt(x^2 + y^2 + z^2)))(dy)/(dt) + (z/(sqrt(x^2 + y^2 + z^2)))(dz)/(dt)`

Hence, evaluating `(ds)/(dt)`  yields what is the relation between `(ds)/(dt)`  and `(dx)/(dt),(dy)/(dt),(dz)/(dt)`  such that: `(ds)/(dt) = (x/(sqrt(x^2 + y^2 + z^2)))(dx)/(dt) + (y/(sqrt(x^2 + y^2 + z^2)))(dy)/(dt) + (z/(sqrt(x^2 + y^2 + z^2)))(dz)/(dt).`

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