We are given the following information about a rectangle:

• The area is $99 \, \text{square units}$.
• The width is $9 \, \text{units}$.

We are asked to find the length of the diagonal, rounding to the nearest tenth of a unit.

Step 1: Find the length of the rectangle

The formula for the area of a rectangle is:

$\text{Area} = \text{Length} \times \text{Width}$

Substitute the given values:

$99 = \text{Length} \times 9$

To find the length, divide both sides by 9:

$\text{Length} = \frac{99}{9} = 11 \, \text{units}$

Step 2: Use the Pythagorean theorem to find the diagonal

The diagonal of a rectangle forms a right triangle with the length and width. According to the Pythagorean theorem:

$\text{Diagonal}^2 = \text{Length}^2 + \text{Width}^2$

Substitute the known values for the length and width:

$\text{Diagonal}^2 = 11^2 + 9^2$ $\text{Diagonal}^2 = 121 + 81$ $\text{Diagonal}^2 = 202$

Now, take the square root of both sides to find the diagonal:

$\text{Diagonal} = \sqrt{202} \approx 14.2 \, \text{units}$

Final Answer:

The length of the diagonal is approximately 14.2 units.

Would you like more details on any part of this solution or have any other questions?

Here are 5 related questions for you to explore:

1. How do you apply the Pythagorean theorem to other geometric shapes?
2. What is the formula for the perimeter of a rectangle?
3. How would the area and diagonal change if the width were doubled?
4. How can you calculate the area of a triangle with a known base and height?
5. How can the Pythagorean theorem be used to find distances in 3D space?

Tip: Always ensure that the units of measurement (like length and width) are consistent before performing any calculations.