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A rectangle has a perimeter of $32\,\text{cm}$ . The length is $6\,\text{cm}$ longer than the width. What is the length?

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Area and perimeter: word problems

Full solution

Q. A rectangle has a perimeter of

32\,\text{cm}

. The length is

6\,\text{cm}

longer than the width. What is the length?

Define Variables: Let's denote the width of the rectangle as $w$ and the length as $l$ . According to the problem, the length is $6\,\text{cm}$ longer than the width, so we can express the length as $l = w + 6$ .
Perimeter Formula: The formula for the perimeter $P$ of a rectangle is $P = 2l + 2w$ . We know the perimeter is $32$ cm, so we can write the equation as $32 = 2(w + 6) + 2w$ .
Expand Equation: Now let's expand the equation: $32 = 2w + 12 + 2w$ .
Combine Like Terms: Combine like terms: $32 = 4w + 12$ .
Subtract $12$ : Subtract $12$ from both sides to solve for $w$ : $32 - 12 = 4w$ .
Divide by $4$ : Perform the subtraction: $20 = 4w$ .
Calculate Width: Divide both sides by $4$ to find $w$ : $\frac{20}{4} = w$ .
Find Length: Calculate the width: $w = 5\,\text{cm}.$
Substitute Width: Now that we have the width, we can find the length using the relationship $l = w + 6$ . Substitute $w = 5\,\text{cm}$ into the equation: $l = 5\,\text{cm} + 6\,\text{cm}$ .
Calculate Length: Calculate the length: $l = 11\,\text{cm}.$

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