Question

The length of a rectangle is five times the breadth. If the minimum perimeter of the rectangle is $180$ cm, then:

• A. Breadth $\leq 15$ cm
• B. Breadth $\geq 15$ cm
• C. Length $\leq 15$ cm
• D. Length = $15$ cm

Answer 1

The Correct Option is B

1. Define variables:

Let $b$ be the breadth of the rectangle (in cm).

The length, $l$, is given as five times the breadth: $l = 5b$.

2. Perimeter formula:

The perimeter, $P$, of a rectangle is given by: $P = 2(l + b)$

3. Substitute and simplify:

Substitute $l = 5b$ into the perimeter formula:

$P = 2(5b + b) = 2(6b) = 12b$

4. Apply the minimum perimeter condition:

The minimum perimeter is given as 180 cm, so $P \geq 180$.

Therefore, $12b \geq 180$

5. Solve for the breadth:

Divide both sides of the inequality by 12:

$b \geq \frac{180}{12} = 15$

6. Interpret the result:

The breadth, $b$, must be greater than or equal to 15 cm.

Correct Answer: (B) Breadth $\geq$ 15 cm

Answer 2

Let the breadth of the rectangle be \( b \) cm and the length be \( l \) cm.
We are given that the length is five times the breadth, so \( l = 5b \).
The perimeter of a rectangle is given by \( P = 2(l + b) \).
Substituting \( l = 5b \), we get:

\[ P = 2(5b + b) = 2(6b) = 12b \]

We are given that the minimum perimeter is 180 cm. Therefore:

\[ 12b \geq 180 \] \[ b \geq \frac{180}{12} \] \[ b \geq 15 \, \text{cm} \]

So the breadth must be greater than or equal to 15 cm.

Therefore, the correct option is (B) Breadth ≥ 15 cm.