The perimeter of a rectangular board is 70 cm. Taking its length as x cm, find its width in terms of x.
If the area of the rectangular board is 300 `cm^(2)`, find its dimensions.

Answer

To solve the problem step by step, we will first find the width of the rectangular board in terms of its length, and then we will find the dimensions of the board given the area.

### Step 1: Understanding the Perimeter
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2(l + b) \]
where \( l \) is the length and \( b \) is the width (or breadth).

### Step 2: Set Up the Equation
Given that the perimeter is 70 cm and the length is \( x \) cm, we can set up the equation:
\[ 2(x + b) = 70 \]

### Step 3: Solve for Width
Dividing both sides by 2:
\[ x + b = 35 \]
Now, we can express the width \( b \) in terms of \( x \):
\[ b = 35 - x \]

### Step 4: Area of the Rectangle
The area \( A \) of a rectangle is given by:
\[ A = l \times b \]
We know the area is 300 cm², so we can write:
\[ x \times (35 - x) = 300 \]

### Step 5: Expand and Rearrange the Equation
Expanding the equation:
\[ 35x - x^2 = 300 \]
Rearranging gives:
\[ x^2 - 35x + 300 = 0 \]

### Step 6: Factor the Quadratic Equation
Now, we need to factor the quadratic equation. We look for two numbers that multiply to 300 and add up to -35. The numbers are -15 and -20:
\[ (x - 15)(x - 20) = 0 \]

### Step 7: Solve for Length
Setting each factor to zero gives us:
1. \( x - 15 = 0 \) → \( x = 15 \)
2. \( x - 20 = 0 \) → \( x = 20 \)

### Step 8: Find the Corresponding Widths
Using the width formula \( b = 35 - x \):
- If \( x = 15 \):
\[ b = 35 - 15 = 20 \]
- If \( x = 20 \):
\[ b = 35 - 20 = 15 \]

### Step 9: Conclusion
The dimensions of the rectangular board can be:
- Length = 20 cm and Width = 15 cm
- Length = 15 cm and Width = 20 cm

Thus, the dimensions of the rectangular board are \( 20 \, \text{cm} \times 15 \, \text{cm} \).

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