Certainly! Let's solve this step-by-step.
### Understanding the Problem
1. Define Variables:
- Let [tex]\( L \)[/tex] be the length of the rectangle.
- Let [tex]\( W \)[/tex] be the width of the rectangle.
- We know that the width is [tex]\( Y \)[/tex] cm less than the length, so [tex]\( W = L - Y \)[/tex].
2. Perimeter Information:
- The perimeter of a rectangle is given by the formula [tex]\( P = 2 \times (L + W) \)[/tex].
- It's given that the perimeter is 64 cm greater than the length. So, [tex]\( P = L + 64 \)[/tex].
### Writing the Perimeter Equation
1. Express the Perimeter with Length and Width:
[tex]\[
P = 2 \times (L + W)
\][/tex]
Substituting [tex]\( W = L - Y \)[/tex]:
[tex]\[
P = 2 \times (L + (L - Y)) = 2 \times (2L - Y) = 4L - 2Y
\][/tex]
2. Relate it to the Given Condition:
Given [tex]\( P = L + 64 \)[/tex]:
[tex]\[
4L - 2Y = L + 64
\][/tex]
### Solving the Equation
1. Isolate the Length [tex]\( L \)[/tex]:
Rearrange the equation:
[tex]\[
4L - L = 64 + 2Y
\][/tex]
Simplify:
[tex]\[
3L = 64 + 2Y
\][/tex]
Solve for [tex]\( L \)[/tex]:
[tex]\[
L = \frac{64 + 2Y}{3}
\][/tex]
2. Given Width Information:
We are informed that the width is [tex]\( Y \)[/tex] cm less than the length; let's assume [tex]\( Y = 5 \)[/tex] cm.
Substitute [tex]\( Y = 5 \)[/tex] into the equation for [tex]\( L \)[/tex]:
[tex]\[
L = \frac{64 + 2 \times 5}{3}
\][/tex]
Solve the expression inside the parentheses first:
[tex]\[
L = \frac{64 + 10}{3} = \frac{74}{3} \approx 24.666666666666668 \, \text{cm}
\][/tex]
### Conclusion
Thus, the length of the rectangle is approximately [tex]\( 24.67 \)[/tex] cm.
