Solving for Rectangle Perimeter Using Area and Ratio

This solution explains how to find the perimeter of a rectangle when given the ratio between its perimeter and breadth, along with its area. We will use basic algebraic methods and geometric formulas to solve this problem.

Problem Breakdown

We are given the following information about a rectangle:

• The ratio of its perimeter (P) to its breadth (b) is 3:1. This can be written as $\frac{P}{b} = \frac{3}{1}$.
• The area (A) of the rectangle is $98 \text{ cm}^2$.

Our goal is to calculate the perimeter of the rectangle.

Key Formulas

We need two main formulas for rectangles:

• Perimeter: $P = 2(l + b)$, where $l$ is the length and $b$ is the breadth.
• Area: $A = l \times b$.

Step-by-Step Solution

1. Using the Perimeter to Breadth Ratio:

   We are given that $\frac{P}{b} = \frac{3}{1}$. This means the perimeter is three times the breadth:

   $P = 3b$

2. Relating Length and Breadth:

   Now, substitute the perimeter formula ($P = 2(l + b)$) into the equation from step 1:

   $2(l + b) = 3b$

   To find the relationship between length ($l$) and breadth ($b$), let's simplify this equation:

   $2l + 2b = 3b$

   Subtract $2b$ from both sides:

   $2l = 3b - 2b$

   $2l = b$

   Now, express the length ($l$) in terms of breadth ($b$):

   $l = \frac{b}{2}$

3. Using the Area Formula:

   We know the area formula is $A = l \times b$. Substitute the expression for $l$ we found in step 2:

   $A = \left(\frac{b}{2}\right) \times b$

   $A = \frac{b^2}{2}$

4. Calculating the Breadth:

   We are given that the area $A = 98 \text{ cm}^2$. Substitute this value into the equation from step 3:

   $98 = \frac{b^2}{2}$

   Multiply both sides by 2 to solve for $b^2$:

   $b^2 = 98 \times 2$

   $b^2 = 196$

   Take the square root of both sides to find the breadth:

   $b = \sqrt{196}$

   $b = 14 \text{ cm}$

5. Calculating the Perimeter:

   Now that we have the breadth ($b = 14 \text{ cm}$), we can find the perimeter using the relationship $P = 3b$ from step 1:

   $P = 3 \times 14$

   $P = 42 \text{ cm}$

Verification

Let's check our answer:

• Breadth $b = 14 \text{ cm}$.
• Length $l = \frac{b}{2} = \frac{14}{2} = 7 \text{ cm}$.
• Perimeter $P = 2(l + b) = 2(7 + 14) = 2(21) = 42 \text{ cm}$.
• Ratio $\frac{P}{b} = \frac{42}{14} = \frac{3}{1}$. This matches the given ratio.
• Area $A = l \times b = 7 \times 14 = 98 \text{ cm}^2$. This matches the given area.

Our calculations are correct.

Final Answer

The perimeter of the rectangle is 42 cm.