Sure, let's solve this step-by-step. Given that we need to draw a rectangle with a perimeter of [tex]\( p = 30 \)[/tex] cm and sides in the ratio [tex]\( 5:3 \)[/tex], we need to find the dimensions of the rectangle. The steps are detailed below:
1. Identify the ratio:
The sides are in the ratio [tex]\(5:3\)[/tex], meaning one side is 5 parts and the other is 3 parts.
2. Sum of the ratio parts:
The total number of parts is [tex]\(5 + 3 = 8\)[/tex].
3. Length of one part of the ratio:
Since the perimeter [tex]\( p \)[/tex] of the rectangle is 30 cm, we know that [tex]\( \text{perimeter} = 2 \times (\text{length} + \text{width}) \)[/tex].
Substituting the ratio dimensions into this:
[tex]\[
p = 2 \times (\text{five parts} + \text{three parts})
\][/tex]
Simplifying further:
[tex]\[
30 = 2 \times (5 + 3) \times \text{part length}
\][/tex]
[tex]\[
30 = 2 \times 8 \times \text{part length}
\][/tex]
[tex]\[
30 = 16 \times \text{part length}
\][/tex]
Solving for the part length:
[tex]\[
\text{part length} = \frac{30}{16} = 1.875 \text{ cm}
\][/tex]
4. Calculate the dimensions of the rectangle:
- Width (5 parts):
[tex]\[
\text{Width} = 5 \times \text{part length} = 5 \times 1.875 = 9.375 \text{ cm}
\][/tex]
- Height (3 parts):
[tex]\[
\text{Height} = 3 \times \text{part length} = 3 \times 1.875 = 5.625 \text{ cm}
\][/tex]
Thus, the rectangle's dimensions are:
- Width: [tex]\(9.375 \, \text{cm}\)[/tex]
- Height: [tex]\(5.625 \, \text{cm}\)[/tex]
These are the dimensions such that the rectangle has a perimeter of 30 cm and the sides are in the ratio [tex]\(5:3\)[/tex].
