# The top and bottom margins of a poster are 4 cm and the side margins are each 6 cm. If the area of printed material on the poster is fixed at 384 square centimeters, how do you find the dimensions of the poster with the smallest area?

Draft

Let a be the width of the poster and b the height.

Let A be the area of the poster to be minimized.

(In this explanation I will omit all the "cm").

As the sum of the illustrated parts:

So let's get a in function of b:

Now,

I have to find the first derivative of the function to minimize it:

The minimum points satisfy the condition

But for obvious reason (b is the height of a poster), b must be positive, so only b=24 is a correct solution.

Now we have to find a:

So, the solution to the problem is

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To find the dimensions of the poster with the smallest area, we can use the fact that the area of printed material is fixed at 384 square centimeters. Let ( x ) represent the width of the printed material and ( y ) represent the height.

The dimensions of the entire poster, including the margins, will be the width of the printed material plus twice the side margins for the total width, and the height of the printed material plus twice the top and bottom margins for the total height.

So, the total width ( W ) of the poster is ( x + 2 \times 6 ) and the total height ( H ) is ( y + 2 \times 4 ).

Since the area of the printed material is fixed at 384 square centimeters, we have the equation:

[ x \times y = 384 ]

Now, we need to express the total area of the poster in terms of ( x ) and ( y ). The area of the entire poster is the product of its width and height:

[ W \times H = (x + 2 \times 6) \times (y + 2 \times 4) ]

We want to minimize this area, so we differentiate it with respect to ( x ) and ( y ), set the derivatives equal to zero, and solve for ( x ) and ( y ).

[ \frac{d(W \times H)}{dx} = \frac{d((x + 2 \times 6) \times (y + 2 \times 4))}{dx} ]

[ \frac{d(W \times H)}{dy} = \frac{d((x + 2 \times 6) \times (y + 2 \times 4))}{dy} ]

After solving these equations, we can find the values of ( x ) and ( y ), which represent the dimensions of the printed material that minimize the area of the entire poster.

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