Use the method of lagrange multipliers to find the dimensions of the rectangle of greatest area that can be inscribed in the ellipse startfraction x squared over 25 endfraction plusstartfraction y squared over 16 endfraction equals1 with sides parallel to the coordinate axes.

Answer with Step-by-step explanation:

Let  a rectangle box whose dimensions are u and v.Then,

must lie on the ellipse

Given equation of ellipse

must lie on the circle therefore,

with

Suppose , we have  to maximize a  function subject to constraints g(u,v)=

Using Lagrange multipliers method

if u=0 then v=0 g(u,v)=

It is absurd condition.

Therefore, we take u and v >0

Substitute the value in ellipse equation then we get

The critical point is (.

Therefore, we concluded that the dimensions of the rectangle of greatest area is attained by choosing a box of dimensions .

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step-by-step explanation:

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step-by-step explanation:

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