# Find the maximum possible total surface area of a cylinder inscribed in a hemisphere of radius 1?

The problem can be stated as a maximization/minimization.

Find maximum/minimum of

subjected to

Introducing a slack variable

The lagrangian is

being analytic, the stationary conditions are given by

giving

Solving for

As we kow the active restriction is

having a minimum

and maximum

The maximum value being

If

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Let a be the inclination to the axis of the cylinder, of the radii of the

hemisphere, to the circular top of the cylinder..Then the radius of the

cylinder is sin a, the height is cos a and the surface area is

As minimum is 0, this a gives the maximum.

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To find the maximum possible total surface area of a cylinder inscribed in a hemisphere of radius 1, we can use calculus. Let ( r ) be the radius of the cylinder and ( h ) be its height.

The total surface area ( A ) of the cylinder is given by:

[ A = 2\pi rh + 2\pi r^2 ]

Given that the cylinder is inscribed in a hemisphere, we have the constraint that ( r \leq 1 ) and ( h \leq 2 ).

To maximize ( A ), we can express ( h ) in terms of ( r ) using the Pythagorean theorem:

[ h = 2\sqrt{1 - r^2} ]

Now, substitute this expression for ( h ) into the formula for ( A ):

[ A(r) = 2\pi r(2\sqrt{1 - r^2}) + 2\pi r^2 ]

To find the maximum surface area, differentiate ( A ) with respect to ( r ), set the derivative equal to zero, and solve for ( r ).

[ \frac{dA}{dr} = 2\pi (2\sqrt{1 - r^2} - \frac{2r^2}{\sqrt{1 - r^2}}) + 4\pi r ]

Setting ( \frac{dA}{dr} = 0 ), we find:

[ 2\sqrt{1 - r^2} - \frac{2r^2}{\sqrt{1 - r^2}} + 2r = 0 ]

Solving this equation yields the value of ( r ). After obtaining ( r ), substitute it back into the expression for ( h ) to find the corresponding value of ( h ). Then calculate the total surface area ( A ) using the formula mentioned earlier.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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