Finding Length and Breadth of a Paper Sheet

The problem asks us to find the length and breadth of a sheet of paper given its area and perimeter. A sheet of paper is typically rectangular in shape.

Let the length of the rectangular sheet be $l$ cm and the breadth be $b$ cm.

We are given the following information:

• Area of the sheet = 240 cm$^2$
• Perimeter of the sheet = 68 cm

The formulas for the area and perimeter of a rectangle are:

• Area = Length $\times$ Breadth ($A = l \times b$)
• Perimeter = 2 $\times$ (Length + Breadth) ($P = 2(l + b)$)

Using the given values, we can set up two equations:

1. $l \times b = 240$
2. $2(l + b) = 68$

Let's simplify the second equation:

$l + b = \frac{68}{2}$

$l + b = 34$

Now we have a system of two equations:

1. $l \times b = 240$
2. $l + b = 34$

We need to find two numbers that multiply to 240 and add up to 34.

We can test the options provided or solve the system algebraically.

Let's check the given options:

As shown in the table, only the dimensions 24 cm and 10 cm satisfy both conditions: the area is 240 cm$^2$ and the perimeter is 68 cm.

Alternatively, we can solve the system of equations algebraically. From $l + b = 34$, we get $l = 34 - b$. Substitute this into the area equation $l \times b = 240$:

$(34 - b) \times b = 240$

$34b - b^2 = 240$

Rearrange the equation into a quadratic form:

$b^2 - 34b + 240 = 0$

We need to find two numbers that multiply to 240 and add up to -34. These numbers are -24 and -10.

So, we can factor the quadratic equation:

$(b - 24)(b - 10) = 0$

This gives two possible values for $b$:

• $b - 24 = 0 \implies b = 24$
• $b - 10 = 0 \implies b = 10$

If $b = 24$ cm, then $l = 34 - b = 34 - 24 = 10$ cm.

If $b = 10$ cm, then $l = 34 - b = 34 - 10 = 24$ cm.

Since length is conventionally greater than or equal to breadth, the length is 24 cm and the breadth is 10 cm. However, the question asks for "length and breadth" and the option lists them in a specific order. The option 24 cm and 10 cm fits our findings.