Length of a rectangle is 3 cm more than width of rectangle and the difference between area of circle and area of rectangle is 84 cm². Fiind the length of rectangle if radius of circle is 7 cm. (circle > rectangle)

Correct Answer: The correct option is 10 cm.

Step-by-Step Explanation:

Step 1: Understand the given information

• Radius of the circle, r = 7 cm

• Let the width of the rectangle be w cm.

• The length of the rectangle is 3 cm more than its width, so length l = (w + 3) cm.

• The area of the circle is greater than the area of the rectangle, and their difference is 84 cm².

Step 2: Calculate the area of the circle

The formula for the area of a circle is:

$\text{Area of circle}=\pi r^2$

Using $\pi =\frac{22}{7}$ and r = 7 cm:

$\text{Area of circle}=\frac{22}{7}\times 7\times 7=22\times 7=154{\text{ cm}}^2$

Step 3: Calculate the area of the rectangle

We are given that:

$\text{Area of circle}-\text{Area of rectangle}=84$

Substituting the area of the circle:

$154-\text{Area of rectangle}=84$

$\text{Area of rectangle}=154-84=70{\text{ cm}}^2$

Step 4: Find the dimensions of the rectangle

The formula for the area of a rectangle is length × width:

$l\times w=70$

Substitute l = w + 3 into the equation:

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

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

Factorizing the quadratic equation:

$w^2+10w-7w-70=0$

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

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

Since width cannot be negative, we take w = 7 cm.

Step 5: Calculate the length of the rectangle

$\text{Length}=w+3=7+3=10\text{ cm}$

Thus, the length of the rectangle is 10 cm.