If length of a rectangle increases by 40% while keeping breadth constant then area of rectangle increased by `24 m^(2)` and perimeter of original rectangle is 32 m. find breadth of rectangle.

Options

A. `8.4 m`
B. 10m
C. 6m
D. 14 m

Answer

To find the breadth of the rectangle, we can follow these steps:

### Step 1: Define the Variables
Let the original length of the rectangle be \( l \) meters and the breadth be \( b \) meters.

### Step 2: Calculate the Area of the Original Rectangle
The area \( A \) of the original rectangle can be calculated as:
\[ A = l \times b \]

### Step 3: Calculate the Increased Length
According to the problem, the length increases by 40%. Therefore, the new length \( l' \) is:
\[ l' = l + 0.4l = 1.4l \]

### Step 4: Calculate the New Area
The area of the rectangle after the increase in length is:
\[ A' = l' \times b = 1.4l \times b = 1.4lb \]

### Step 5: Find the Increase in Area
The increase in area is given as \( 24 \, m^2 \). Therefore, we can set up the equation:
\[ A' - A = 24 \]
Substituting the areas we calculated:
\[ 1.4lb - lb = 24 \]
This simplifies to:
\[ 0.4lb = 24 \]

### Step 6: Solve for \( lb \)
Now, we can solve for \( lb \):
\[ lb = \frac{24}{0.4} = 60 \]

### Step 7: Use the Perimeter to Find Another Equation
The perimeter \( P \) of the rectangle is given as \( 32 \, m \). The formula for the perimeter is:
\[ P = 2(l + b) = 32 \]
Dividing both sides by 2 gives:
\[ l + b = 16 \]

### Step 8: Set Up the System of Equations
Now we have two equations:
1. \( lb = 60 \) (Equation 1)
2. \( l + b = 16 \) (Equation 2)

### Step 9: Solve the System of Equations
From Equation 2, we can express \( l \) in terms of \( b \):
\[ l = 16 - b \]

Substituting this into Equation 1:
\[ (16 - b)b = 60 \]
This expands to:
\[ 16b - b^2 = 60 \]
Rearranging gives:
\[ b^2 - 16b + 60 = 0 \]

### Step 10: Solve the Quadratic Equation
We can use the quadratic formula \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \) where \( A = 1, B = -16, C = 60 \):
\[ b = \frac{16 \pm \sqrt{(-16)^2 - 4 \cdot 1 \cdot 60}}{2 \cdot 1} \]
Calculating the discriminant:
\[ b = \frac{16 \pm \sqrt{256 - 240}}{2} = \frac{16 \pm \sqrt{16}}{2} = \frac{16 \pm 4}{2} \]
This gives us two possible values for \( b \):
\[ b = \frac{20}{2} = 10 \quad \text{or} \quad b = \frac{12}{2} = 6 \]

### Step 11: Find the Valid Breadth
Since both values are valid, we can substitute back to find the corresponding lengths:
1. If \( b = 10 \), then \( l = 16 - 10 = 6 \) (not valid since \( lb \neq 60 \)).
2. If \( b = 6 \), then \( l = 16 - 6 = 10 \) (valid since \( 10 \times 6 = 60 \)).

Thus, the breadth of the rectangle is:
\[ \boxed{6 \, m} \]