What is the greatest area (in sq. units) of a rectangle the sum of whose 3 sides is equal = 100.

Options

A. 625
B. 1250
C. 883.33
D. 666.66

Answer

To find the greatest area of a rectangle given that the sum of its three sides equals 100, we can follow these steps:

### Step 1: Define the variables
Let the length of the rectangle be \( L \) and the breadth be \( B \).

### Step 2: Set up the equation based on the problem
According to the problem, the sum of three sides of the rectangle is given by:
\[ L + 2B = 100 \]

### Step 3: Express \( L \) in terms of \( B \)
From the equation \( L + 2B = 100 \), we can express \( L \) as:
\[ L = 100 - 2B \]

### Step 4: Write the area of the rectangle
The area \( A \) of the rectangle is given by:
\[ A = L \times B \]
Substituting the expression for \( L \):
\[ A = (100 - 2B) \times B \]
This simplifies to:
\[ A = 100B - 2B^2 \]

### Step 5: Find the maximum area using calculus
To find the value of \( B \) that maximizes the area, we need to take the derivative of \( A \) with respect to \( B \) and set it to zero:
\[ \frac{dA}{dB} = 100 - 4B \]
Setting the derivative equal to zero for maximization:
\[ 100 - 4B = 0 \]
Solving for \( B \):
\[ 4B = 100 \implies B = 25 \]

### Step 6: Find the corresponding length \( L \)
Now, substituting \( B = 25 \) back into the equation for \( L \):
\[ L = 100 - 2(25) = 100 - 50 = 50 \]

### Step 7: Calculate the maximum area
Now we can calculate the maximum area:
\[ A = L \times B = 50 \times 25 = 1250 \]

### Conclusion
The greatest area of the rectangle is:
\[ \boxed{1250} \text{ square units} \]

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