To solve the problem of finding the length and width of the rectangle given that its diagonal is [tex]\(\sqrt{1664}\)[/tex] cm and the length is 5 times the width, we can follow these steps step by step.
1. Understand the Problem:
- Diagonal ([tex]\(d\)[/tex]) = [tex]\(\sqrt{1664}\)[/tex] cm.
- Let the width be [tex]\(w\)[/tex] cm.
- The length ([tex]\(l\)[/tex]) is given as 5 times the width, so [tex]\(l = 5w\)[/tex].
2. Apply the Pythagorean Theorem:
- In a rectangle, the relationship between the length, width, and diagonal is given by the Pythagorean theorem:
[tex]\[
d^2 = l^2 + w^2
\][/tex]
- Substituting the given values:
[tex]\[
(\sqrt{1664})^2 = (5w)^2 + w^2
\][/tex]
- Simplify the equation:
[tex]\[
1664 = 25w^2 + w^2
\][/tex]
[tex]\[
1664 = 26w^2
\][/tex]
3. Solve for [tex]\(w\)[/tex]:
- Isolate [tex]\(w^2\)[/tex]:
[tex]\[
w^2 = \frac{1664}{26}
\][/tex]
- Calculate [tex]\(w^2\)[/tex]:
[tex]\[
w^2 = 64
\][/tex]
- Take the square root of both sides to find [tex]\(w\)[/tex]:
[tex]\[
w = \sqrt{64} = 8 \, \text{cm}
\][/tex]
4. Find the Length [tex]\(l\)[/tex]:
- Since [tex]\(l = 5w\)[/tex]:
[tex]\[
l = 5 \times 8 = 40 \, \text{cm}
\][/tex]
5. Confirm the Diagonal:
- Verify by recalculating the diagonal using the found values:
[tex]\[
d = \sqrt{l^2 + w^2} = \sqrt{40^2 + 8^2} = \sqrt{1600 + 64} = \sqrt{1664} \, \text{cm}
\][/tex]
- The calculated diagonal matches the given diagonal, confirming our solution is correct.
So, the width of the rectangle is [tex]\(\boxed{8 \, \text{cm}}\)[/tex] and the length is [tex]\(\boxed{40 \, \text{cm}}\)[/tex].
