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39. The perimeter of a rectangle is 40 m. The length of the rectangle is 4 m less than 5 times its breadth. Find the length of the rectangle.

Sagot :

Certainly! Let's solve this step-by-step.

1. Define the variables:
- Let [tex]\( l \)[/tex] be the length of the rectangle.
- Let [tex]\( b \)[/tex] be the breadth of the rectangle.

2. Given conditions:
- The perimeter of the rectangle is [tex]\( 40 \)[/tex] meters.
- The length ([tex]\( l \)[/tex]) is 4 meters less than 5 times the breadth ([tex]\( b \)[/tex]).

3. Formulate the equations:
- The formula for the perimeter of a rectangle is [tex]\( P = 2(l + b) \)[/tex].
- Given [tex]\( P = 40 \)[/tex], we have:
[tex]\[ 2(l + b) = 40 \][/tex]

4. Express the length in terms of the breadth:
- According to the problem, [tex]\( l = 5b - 4 \)[/tex].

5. Substitute [tex]\( l = 5b - 4 \)[/tex] into the perimeter equation:
[tex]\[ 2((5b - 4) + b) = 40 \][/tex]

6. Simplify the equation:
[tex]\[ 2(6b - 4) = 40 \][/tex]

7. Distribute the 2 over the terms inside the parenthesis:
[tex]\[ 12b - 8 = 40 \][/tex]

8. Solve for [tex]\( b \)[/tex]:
- Add 8 to both sides of the equation:
[tex]\[ 12b = 48 \][/tex]
- Divide both sides by 12:
[tex]\[ b = 4 \][/tex]

9. Find the length using the breadth:
- Substitute [tex]\( b = 4 \)[/tex] back into the equation [tex]\( l = 5b - 4 \)[/tex]:
[tex]\[ l = 5(4) - 4 \][/tex]
[tex]\[ l = 20 - 4 \][/tex]
[tex]\[ l = 16 \][/tex]

10. Conclusion:
- The length of the rectangle is [tex]\( 16 \)[/tex] meters.