Question

The top and bottom margins of a poster are each 6 cm and the side margins are each 4 cm. If the area of printed material on the poster is fixed at \({\bf{384}}\;{\bf{c}}{{\bf{m}}^{\bf{2}}}\), find the dimensions of the poster with the smallest area.

Short answer

The dimensions of the poster are 24 cm and 36 cm, respectively.

Step-by-step solution

Step 1-Find the function for the area of the window

The figure below represents the sketch of the poster.

The area of the rectangle inside is:

\(\begin{aligned}{c}xy = 384\\y = \frac{{384}}{x}\end{aligned}\)

The total area of the poster is given as:

\(\begin{aligned}{c}A\left( x \right) = \left( {8 + x} \right)\left( {12 + \frac{{384}}{x}} \right)\\ = 12\left( {8 + x} \right)\left( {1 + \frac{{32}}{x}} \right)\\ = 12\left( {8 + x + \frac{{256}}{x} + 32} \right)\\ = 12\left( {40 + x + \frac{{256}}{x}} \right)\end{aligned}\)

Step 2-Differentiate the function of the area of the poster

Differentiate the function \(A\left( x \right) = 12\left( {40 + x + \frac{{256}}{x}} \right)\).

\(\begin{aligned}{c}A'\left( x \right) = \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {12\left( {40 + x + \frac{{256}}{x}} \right)} \right)\\ = 12\left( {0 + 1 - \frac{{256}}{{{x^2}}}} \right)\\ = 12\left( {1 - \frac{{256}}{{{x^2}}}} \right)\end{aligned}\)

Differentiate the function \(A'\left( x \right)\).

\(\begin{aligned}{c}A''\left( x \right) = \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {12\left( {1 - \frac{{256}}{{{x^2}}}} \right)} \right)\\ = 12\left( {0 - \frac{{512}}{{{x^3}}}} \right)\end{aligned}\)

Step 3-Find the values of x at which total area of poster is maximum

Find the roots of \(A'\left( x \right) = 0\).

\(\begin{aligned}{c}12\left( {1 - \frac{{256}}{{{x^2}}}} \right) = 0\\{x^2} = 256\\x = 16\end{aligned}\)

As \(A''\left( x \right) < 0\), therefore the area of the window is maximum for \(x = 16\).

Step 4-Find the value of y

Substitute 16 for x in the equation \(y = \frac{{384}}{x}\).

\(\begin{aligned}{c}y = \frac{{384}}{{16}}\\ = 24\end{aligned}\)

Now, the dimensions are shown below:

\(\begin{aligned}{c}x + 8 = 16 + 8\\ = 24\end{aligned}\)

And,

\(\begin{aligned}{c}y + 12 = 24 + 12\\ = 36\end{aligned}\)

The dimensions of the poster are 24 cm and 36 cm, respectively