If x space a n d space y are the dimensions of a rectangle whose area is 27 and whose perimeter is as small as possible, then x squared plus y squared equals
	a.	
50
	b.	
60
	c.	
44
	d.	
54
	e.	
32


To solve this problem, we need to determine the dimensions \( x \) and \( y \) of a rectangle such that the area is 27 and the perimeter is minimized. Afterward, we'll calculate \( x^2 + y^2 \).

### Step 1: Define the equations

- The area of the rectangle is given by:
  \[
  x \cdot y = 27
  \]
  
- The perimeter of the rectangle is:
  \[
  P = 2(x + y)
  \]
  
We want to minimize the perimeter.

### Step 2: Minimize the perimeter
To minimize \( P = 2(x + y) \), express \( y \) in terms of \( x \) using the area equation:
\[
y = \frac{27}{x}
\]
Substitute this into the perimeter equation:
\[
P = 2\left(x + \frac{27}{x}\right)
\]
To minimize this, we take the derivative of \( P \) with respect to \( x \) and set it equal to zero:
\[
\frac{dP}{dx} = 2\left(1 - \frac{27}{x^2}\right) = 0
\]
This implies:
\[
1 = \frac{27}{x^2} \quad \Rightarrow \quad x^2 = 27 \quad \Rightarrow \quad x = \sqrt{27} = 3\sqrt{3}
\]
Since \( y = \frac{27}{x} \):
\[
y = \frac{27}{3\sqrt{3}} = 3\sqrt{3}
\]
Thus, \( x = y = 3\sqrt{3} \), which means the rectangle is a square.

### Step 3: Calculate \( x^2 + y^2 \)
\[
x^2 + y^2 = (3\sqrt{3})^2 + (3\sqrt{3})^2 = 27 + 27 = 54
\]

### Final Answer:
The correct answer is \( \boxed{54} \) (Option D).

Would you like more details on this problem or have any questions?

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Here are 8 related questions:

1. How do you find the derivative of a function?
2. What are the properties of a rectangle that minimize its perimeter?
3. Can the dimensions of a rectangle with a fixed area ever be negative?
4. What is the relationship between the area and perimeter of a square?
5. How does calculus help in optimizing functions?
6. Why do we set the derivative equal to zero to find the minimum or maximum value?
7. How do you solve for variables when given two simultaneous equations?
8. What other methods can be used to minimize or maximize a perimeter or area?

**Tip:** When minimizing or maximizing a function, always check the second derivative to confirm if it is a minimum or maximum.

Learn how to minimize the perimeter of a rectangle with an area of 27 units. Understand the relationship between area and perimeter in geometry with step-by-step solutions and calculations. Suitable for high school students.

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