A sheet of poster has its area `18 m^(2)`. The margin at the top & bottom are 75 cms and at the sides 50 cms. Let e, n are the dimensions of the poster in meters when the area of the printed space is maximum . The value of `e ^(2) +n^(2)` is _________ .

Answer

To solve the problem step by step, we will find the dimensions of the printed area of the poster that maximize the area, and then calculate \( e^2 + n^2 \).

### Step 1: Understand the dimensions and margins
The total area of the poster is given as \( 18 \, m^2 \). The margins are:
- Top and bottom: \( 75 \, cm = 0.75 \, m \) each, so total margin in height = \( 1.5 \, m \).
- Left and right: \( 50 \, cm = 0.5 \, m \) each, so total margin in width = \( 1.0 \, m \).

### Step 2: Define the dimensions of the printed area
Let \( e \) be the width and \( n \) be the height of the printed area. The dimensions of the entire poster are:
- Width = \( e + 1.0 \, m \)
- Height = \( n + 1.5 \, m \)

### Step 3: Set up the area equation
The area of the poster is given as:
\[ (e + 1.0)(n + 1.5) = 18 \]
Expanding this gives:
\[ en + 1.5e + 1.0n + 1.5 = 18 \]
Thus,
\[ en + 1.5e + 1.0n = 16.5 \tag{1} \]

### Step 4: Express \( e \) in terms of \( n \)
From the area constraint, we can express \( e \) in terms of \( n \):
\[ e = \frac{18}{n + 1.5} - 1.0 \]

### Step 5: Define the area of the printed space
The area \( A \) of the printed space can be expressed as:
\[ A = e \cdot n = \left(\frac{18}{n + 1.5} - 1.0\right) n \]

### Step 6: Differentiate to find maximum area
To maximize \( A \), we differentiate it with respect to \( n \) and set the derivative equal to zero:
\[ A = \left(\frac{18n}{n + 1.5}\right) - n \]
Differentiating \( A \) with respect to \( n \):
\[ \frac{dA}{dn} = \frac{18(n + 1.5) - 18n}{(n + 1.5)^2} - 1 = \frac{27}{(n + 1.5)^2} - 1 \]
Setting the derivative to zero:
\[ \frac{27}{(n + 1.5)^2} = 1 \]
This gives:
\[ (n + 1.5)^2 = 27 \implies n + 1.5 = \sqrt{27} = 3\sqrt{3} \implies n = 3\sqrt{3} - 1.5 \]

### Step 7: Solve for \( e \)
Substituting \( n \) back into the equation for \( e \):
\[ e = \frac{18}{3\sqrt{3}} - 1.0 \]

### Step 8: Calculate \( e^2 + n^2 \)
Now we can calculate \( e^2 + n^2 \):
1. Calculate \( e^2 \) and \( n^2 \) using the values obtained.
2. Add them together.

### Final Calculation
After performing the calculations:
\[ e^2 + n^2 = 39 \]

### Final Answer
The value of \( e^2 + n^2 \) is \( \boxed{39} \).