To find the length of the diagonal of the rectangle, we can follow these steps:
1. Calculate the width of the rectangle:
- The area of a rectangle is given by the formula:
[tex]\[
\text{Area} = \text{Length} \times \text{Width}
\][/tex]
- Here, the area is 135 square units and the length is 9 units.
- Therefore, we can determine the width by rearranging the formula to solve for width:
[tex]\[
\text{Width} = \frac{\text{Area}}{\text{Length}} = \frac{135}{9} = 15 \text{ units}
\][/tex]
2. Apply the Pythagorean theorem:
- To find the diagonal of the rectangle, we treat the length, width, and diagonal as sides of a right triangle. The formula from the Pythagorean theorem is:
[tex]\[
c = \sqrt{a^2 + b^2}
\][/tex]
where [tex]\(c\)[/tex] is the hypotenuse (diagonal), and [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the legs (length and width of the rectangle).
- Here, [tex]\(a = 9\)[/tex] units (length) and [tex]\(b = 15\)[/tex] units (width).
3. Calculate the diagonal:
- Plugging in the values:
[tex]\[
c = \sqrt{9^2 + 15^2} = \sqrt{81 + 225} = \sqrt{306}
\][/tex]
- Now, compute the square root of 306:
[tex]\[
\sqrt{306} \approx 17.4928556845359
\][/tex]
4. Round the result to the nearest tenth:
- Rounding 17.4928556845359 to the nearest tenth:
[tex]\[
17.5 \text{ units}
\][/tex]
Therefore, the length of the diagonal of the rectangle is approximately [tex]\(17.5\)[/tex] units.
