This problem requires us to find the length of a rectangle using information about its dimensions and the area difference between it and a circle.

Calculating the Circle's Area

We are given that the radius of the circle, denoted by $r$, is 7 cm. The formula for the area of a circle ($A_{\text{circle}}$) is given by:

$$ A_{\text{circle}} = \pi r^2 $$

Using the approximation $\pi \approx \frac{22}{7}$, we can calculate the area:

$$ A_{\text{circle}} = \frac{22}{7} \times (7 \text{ cm})^2 $$

$$ A_{\text{circle}} = \frac{22}{7} \times 49 \text{ cm}^2 $$

$$ A_{\text{circle}} = 22 \times 7 \text{ cm}^2 $$

$$ A_{\text{circle}} = 154 \text{ cm}^2 $$

Determining the Rectangle's Area

The problem states that the difference between the area of the circle and the area of the rectangle ($A_{\text{rect}}$) is 84 cm². Since the circle's area is greater than the rectangle's area (circle > rectangle), we can write this relationship as:

$$ A_{\text{circle}} - A_{\text{rect}} = 84 \text{ cm}^2 $$

Now, we substitute the calculated area of the circle:

$$ 154 \text{ cm}^2 - A_{\text{rect}} = 84 \text{ cm}^2 $$

To find the area of the rectangle, we rearrange the equation:

$$ A_{\text{rect}} = 154 \text{ cm}^2 - 84 \text{ cm}^2 $$

$$ A_{\text{rect}} = 70 \text{ cm}^2 $$

Relating Rectangle Dimensions and Solving for Width

Let the width of the rectangle be $W$ cm and the length be $L$ cm. We are given two conditions:

• The length is 3 cm more than the width: $L = W + 3$.
• The area of the rectangle is 70 cm²: $A_{\text{rect}} = L \times W = 70$.

We can substitute the first condition into the second condition:

$$ (W + 3) \times W = 70 $$

Expanding this equation gives us a quadratic equation:

$$ W^2 + 3W = 70 $$

Rearranging the terms to form a standard quadratic equation:

$$ W^2 + 3W - 70 = 0 $$

To solve for $W$, we can factor the quadratic equation. We need two numbers that multiply to -70 and add up to 3. These numbers are 10 and -7.

$$ (W + 10)(W - 7) = 0 $$

This gives two possible values for $W$:

• $W + 10 = 0 \implies W = -10$
• $W - 7 = 0 \implies W = 7$

Since the width of a rectangle cannot be negative, we must have $W = 7$ cm.

Calculating the Rectangle's Length

Now that we have found the width ($W = 7$ cm), we can find the length using the relationship $L = W + 3$:

$$ L = 7 \text{ cm} + 3 \text{ cm} $$

$$ L = 10 \text{ cm} $$

Final Check

If the width is 7 cm and the length is 10 cm:

• The length is indeed 3 cm more than the width ($10 = 7 + 3$).
• The area of the rectangle is $10 \text{ cm} \times 7 \text{ cm} = 70 \text{ cm}^2$.
• The difference between the circle's area ($154 \text{ cm}^2$) and the rectangle's area ($70 \text{ cm}^2$) is $154 - 70 = 84 \text{ cm}^2$, which matches the given condition.

Therefore, the length of the rectangle is 10 cm.