Question

the top and bottom margins of a poster are each 12 cm and the side margins are each 8 cm. if the area of printed material on the poster is fixed at 1,536 cm², find the dimensions (in cm) of the poster with the smallest area.
width
height
cm
cm

Explanation:

Step 1: Define variables

Let the height of the printed area be \(x\) cm and the width of the printed area be \(y\) cm. Then \(xy = 1536\), so \(y=\frac{1536}{x}\).
The height of the poster \(H=x + 24\) (since top and bottom margins are \(12\) cm each) and the width of the poster \(W=y + 16\) (since side margins are \(8\) cm each).
The area of the poster \(A=(x + 24)(y+16)\).
Substitute \(y = \frac{1536}{x}\) into the area formula: \(A=(x + 24)(\frac{1536}{x}+16)\).
Expand the formula:

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Step 2: Find the derivative of \(A\)

Differentiate \(A(x)\) with respect to \(x\). Using the power rule \((u + v+w)'=u'+v'+w'\), where \(u = 1920\) (\(u'=0\)), \(v = 16x\) (\(v'=16\)), \(w=\frac{36864}{x}=36864x^{-1}\) (\(w'=- 36864x^{-2}\)).
So \(A'(x)=16-\frac{36864}{x^{2}}\).

Step 3: Set the derivative equal to zero

Set \(A'(x)=0\):

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We take \(x = 48\) (since \(x>0\)).

Step 4: Find \(y\)

Since \(y=\frac{1536}{x}\), when \(x = 48\), \(y=\frac{1536}{48}=32\).

Step 5: Find the dimensions of the poster

The height of the poster \(H=x + 24=48+24 = 72\) cm.
The width of the poster \(W=y + 16=32+16=48\) cm.

Answer:

height \(72\) cm, width \(48\) cm