1. Suppose an airline policy states that the sum of the length, width, and height of a carry-on suitcase should not exceed 45 inches. What are the dimensions of the largest carry-on suitcase whose height is twice its width?

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2. Find the coordinates of the point P on the parabola given by \( y= 1-x^2 \) that is closest to the point \( (2,\frac{1}{2})\).

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3. A rectangle is constructed so that one side on the x-axis, and two of its vertices are on the parabola given by \( y=16-x^2\). What are the dimensions of the rectangle with maximum area? What is that area?

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4. A cylindrical soda can is constructed out of aluminum so that the the thickness of the top and bottom is twice the thickness of the side. Find the dimensions of a soda can whose volume is 354 \( cm^3\) which uses as little aluminum as possible.

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5. A rectangular flower garden with an area of \( 30 m^2 \) is surrounded by a rectangular grass border \( 1m\) wide on the left and right sides of the garden and \( 2 m\) wide on the other two sides. Find the dimensions of the garden that will minimize the combined area of the garden and the border.

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