The perimeter and area of a rectangle are 32 and 60 respectively. What is its breadth?

Options

A. 10
B. 16
C. 6
D. 30

Explanation

To find the breadth of the rectangle given its perimeter and area, we can follow these steps:

### Step 1: Write down the formulas for perimeter and area of a rectangle.
The perimeter \( P \) of a rectangle is given by:
\[ P = 2 \times (l + b) \]
where \( l \) is the length and \( b \) is the breadth.

The area \( A \) of a rectangle is given by:
\[ A = l \times b \]

### Step 2: Set up the equations using the given values.
From the problem, we know:
- The perimeter \( P = 32 \)
- The area \( A = 60 \)

Using the perimeter formula:
\[ 2 \times (l + b) = 32 \]
Dividing both sides by 2:
\[ l + b = 16 \quad \text{(Equation 1)} \]

Using the area formula:
\[ l \times b = 60 \quad \text{(Equation 2)} \]

### Step 3: Express one variable in terms of the other.
From Equation 1, we can express \( l \) in terms of \( b \):
\[ l = 16 - b \]

### Step 4: Substitute this expression into the area equation.
Substituting \( l \) in Equation 2:
\[ (16 - b) \times b = 60 \]
Expanding this gives:
\[ 16b - b^2 = 60 \]

### Step 5: Rearrange the equation into standard quadratic form.
Rearranging the equation:
\[ b^2 - 16b + 60 = 0 \]

### Step 6: Solve the quadratic equation using the quadratic formula.
The quadratic formula is given by:
\[ b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Here, \( A = 1 \), \( B = -16 \), and \( C = 60 \).

Calculating the discriminant:
\[ B^2 - 4AC = (-16)^2 - 4 \times 1 \times 60 = 256 - 240 = 16 \]

Now, substituting into the quadratic formula:
\[ b = \frac{16 \pm \sqrt{16}}{2 \times 1} = \frac{16 \pm 4}{2} \]

Calculating the two possible values for \( b \):
1. \( b = \frac{20}{2} = 10 \)
2. \( b = \frac{12}{2} = 6 \)

### Step 7: Determine the breadth.
Since breadth cannot be greater than the length, we take \( b = 6 \) as the breadth.

### Final Answer:
The breadth of the rectangle is \( 6 \).

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