Length of rectangle is 20% more than its width. Find the diagonal of rectangle if area of rectangle is 4320 `cm^(2).`

Options

A. A)`15 sqrt17 cm`
B. B)`12 sqrt61 cm`
C. C)`14 sqrt21 cm`
D. D)`10 sqrt21 cm`

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Define Variables
Let the width of the rectangle be \( w \) cm. According to the problem, the length \( l \) is 20% more than the width. Therefore, we can express the length as:
\[ l = w + 0.2w = 1.2w \]

### Step 2: Use the Area Formula
The area \( A \) of the rectangle is given by the formula:
\[ A = l \times w \]
We know from the problem that the area is \( 4320 \, \text{cm}^2 \). Substituting the expression for length, we get:
\[ 4320 = (1.2w) \times w \]
This simplifies to:
\[ 4320 = 1.2w^2 \]

### Step 3: Solve for Width
To find \( w^2 \), we can rearrange the equation:
\[ w^2 = \frac{4320}{1.2} \]
Calculating the right side:
\[ w^2 = 3600 \]
Now, taking the square root of both sides:
\[ w = \sqrt{3600} = 60 \, \text{cm} \]

### Step 4: Calculate Length
Now that we have the width, we can find the length:
\[ l = 1.2w = 1.2 \times 60 = 72 \, \text{cm} \]

### Step 5: Calculate the Diagonal
The diagonal \( d \) of the rectangle can be calculated using the Pythagorean theorem:
\[ d = \sqrt{l^2 + w^2} \]
Substituting the values we found:
\[ d = \sqrt{72^2 + 60^2} \]
Calculating the squares:
\[ d = \sqrt{5184 + 3600} = \sqrt{8784} \]

### Step 6: Simplify the Diagonal
Now we can simplify \( \sqrt{8784} \):
\[ d = \sqrt{144 \times 61} = 12\sqrt{61} \, \text{cm} \]

### Final Answer
Thus, the diagonal of the rectangle is:
\[ d = 12\sqrt{61} \, \text{cm} \]

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