To understand the problem, let’s assume that we have a rectangular pen and that [tex]\( a \)[/tex] represents the length of the pen. We need to determine the width of the pen and calculate the perimeter of the rectangle.
1. Width Calculation:
- According to the question, the width of the pen is half of its length. Therefore, if [tex]\( a \)[/tex] is the length of the pen, the width [tex]\( w \)[/tex] can be calculated as:
[tex]\[
w = \frac{a}{2}
\][/tex]
2. Perimeter Calculation:
- The formula for the perimeter [tex]\( P \)[/tex] of a rectangle is given by:
[tex]\[
P = 2 \times (\text{length} + \text{width})
\][/tex]
- Substituting the values of length [tex]\( a \)[/tex] and width [tex]\( w \)[/tex] into the formula, we get:
[tex]\[
P = 2 \times \left(a + \frac{a}{2}\right)
\][/tex]
- Simplify the expression inside the parentheses:
[tex]\[
a + \frac{a}{2} = \frac{2a}{2} + \frac{a}{2} = \frac{3a}{2}
\][/tex]
- Substituting this back into the perimeter formula:
[tex]\[
P = 2 \times \frac{3a}{2}
\][/tex]
- The twos cancel out in the calculation, leaving:
[tex]\[
P = 3a
\][/tex]
Given that the length [tex]\( a \)[/tex] is assumed to be 30, let's plug in this value:
1. Width:
[tex]\[
w = \frac{30}{2} = 15
\][/tex]
2. Perimeter:
[tex]\[
P = 3 \times 30 = 90
\][/tex]
Therefore, if the length of the pen [tex]\( a \)[/tex] is 30, the width of the pen is 15, and the perimeter of the rectangular pen is 90. The final results are:
- The width is [tex]\( 15 \)[/tex] units.
- The perimeter is [tex]\( 90 \)[/tex] units.
