What is the answer to a small box holds 75% as much as a large box if the small box holds 36 units how much does a large box hol
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The distance between a point 
on the given plane and the point (0, 2, 4) is
but since
and 
share critical points, we can instead consider the problem of optimizing subject to 
.
The Lagrangian is
with partial derivatives (set equal to 0)
Solve for
:
which gives the critical point
We can confirm that this is a minimum by checking the Hessian matrix of:
is positive definite (we see its determinant and the determinants of its leading principal minors are positive), which indicates that there is a minimum at this critical point.
At this point, we get a distance from (0, 2, 4) of
but since
The Lagrangian is
with partial derivatives (set equal to 0)
Solve for
which gives the critical point
We can confirm that this is a minimum by checking the Hessian matrix of
At this point, we get a distance from (0, 2, 4) of
Step-by-step explanation:
First factor out the negative sign from the expression and reorder the terms
That's
<u>Using trigonometric </u><u>identities</u>
That's
<h3><u>Rewrite the expression</u>
That's
We have
<h3><u>Rewrite the second fraction</u>
That's
<h3>Since they have the same denominator we can write the fraction as
Using the identity
<h3><u>Rewrite the expression</u>
We have
<h3><u>Using the trigonometric identity</u>
<h3><u>Rewrite the expression</u>
That's
<h3>Which is
<h3><u>Using the trigonometric identity</u>
<h3>Rewrite the expression
That's
<h3><u>Simplify the expression using symmetry of trigonometric functions</u>
That's
<h3><u>Remove the parenthesis </u>
We have the final answer as
<h2>As proven
Hope this helps you