Sure, let's solve this problem step-by-step.
1. Define the variables:
- Let [tex]\( L \)[/tex] represent the length of the rectangle.
- Let [tex]\( W \)[/tex] represent the width of the rectangle.
2. Set up the equations:
- We are told that the length is 4 centimeters less than twice the width. This translates to the equation:
[tex]\[ L = 2W - 4 \][/tex]
- We are also told that the perimeter of the rectangle is 34 centimeters. The formula for the perimeter of a rectangle is:
[tex]\[ \text{Perimeter} = 2L + 2W \][/tex]
Substituting the given perimeter:
[tex]\[ 2L + 2W = 34 \][/tex]
3. Substitute [tex]\( L \)[/tex] from the first equation into the second equation:
- From the first equation, we have [tex]\( L = 2W - 4 \)[/tex]. Substitute this into the perimeter equation:
[tex]\[ 2(2W - 4) + 2W = 34 \][/tex]
4. Solve for [tex]\( W \)[/tex]:
- Expand and simplify the equation:
[tex]\[ 4W - 8 + 2W = 34 \][/tex]
Combine like terms:
[tex]\[ 6W - 8 = 34 \][/tex]
Add 8 to both sides of the equation:
[tex]\[ 6W = 42 \][/tex]
Divide by 6:
[tex]\[ W = 7 \][/tex]
5. Solve for [tex]\( L \)[/tex]:
- Now, use the value of [tex]\( W \)[/tex] in the first equation [tex]\( L = 2W - 4 \)[/tex]:
[tex]\[ L = 2(7) - 4 \][/tex]
[tex]\[ L = 14 - 4 \][/tex]
[tex]\[ L = 10 \][/tex]
6. Conclusion:
- The length [tex]\( L \)[/tex] is 10 cm.
- The width [tex]\( W \)[/tex] is 7 cm.
Thus, the dimensions of the rectangle are:
[tex]\[ \text{Length} = 10 \text{ cm}, \text{ Width} = 7 \text{ cm} \][/tex]
The correct answer is:
[tex]\[ \boxed{\text{length } 10 \text{ cm; width } 7 \text{ cm}} \][/tex]
