The perimeter of a standard-sized rectangular rug is 40 ft. The length is 2 ft longer than the width. Find the dimensions.

Question

College Algebra

Answer 1

Strategy: First, let us put into equations what we are given in words. Then later we will plug in values and solve for one of our unknowns. (summarize: equation setup, plug in values and solve for one unknown, find other unknowns.)

Step 1: Perimeter= sum of all sides of the object. 

For a rectangle, the two opposite sides are congruent. i.e. the 2 lengths are same length, and the 2 widths are same length.  We can call them l for the length, and w for the width.

Step 2: Perimeter = 2l + 2w (since the sum L + L + W + W = Perimeter)

Step 3: We are told l = 2 + w (since we are told the length is 2 feet longer than the width)

Step 4: We are told Perimeter = 40

Step 5: Now let's use the above information to plug in and solve for w.

We know l = 2 + w from step 3

So plug this into step 2, and get: Perimeter = 2(2 + w) + 2w

We also know Perimeter=40 from step 4, so plug this in and get, 40 = 2(2 + w) + 2w

Let's solve for w now:

use distributive property: 40 = 4 + 2w +2w

combine like terms: 40 = 4 + 4w

subtract 4 from both sides, so we can isolate the w: 36 = 4w

divide by 4 on both sides, so we can isolate w: 9 = w

include the units: so w = 9 ft
Step 6: Lets find l now, by plugging in w =9 into step 3, l = 2 + w

so l = 2 + 9 = 11

so l = 11 ft

Solution: The dimensions are: width = 9 ft, length = 11 ft.

Check our Answer: To check our answer, let's plug all known values into equation in step 2, Perimeter = 2l + 2w
since perimeter is 40, l is 11, and w is 9, we get: 40=2*11 + 2*9
simplifying by multiplication, we get: 40=22+18
simplifying by multiplication, we get: 40=40 . This is what we want to happen! Both sides are equal.

 

Since both sides of our equation are equal, i.e. 40 = 40, we are assured that values we found for l and w are accurate.

So the dimensions for the rectangle are truely, 9ft width x 11ft length.

Answer 2

For this answer let  l=length w=width and p=perimeter

 

The formula for the perimeter of a rectangle is p=2l+2w (we will call this formula 1)

 

From the question, we know that p=40ft

 

Step One: From the question, we know that the length of the rectangle is 2ft plus the measurement of the width, so we can create the following equation

 

l=w+2 

 

 

Step Two: Now we will plug in l from the equation from step one into formula 1 using substitution and solve for w.

 

p=2l+2w  (formula 1)

 

p=2(w+2)+2w   (found by plugging w+2 in for l)

 

p=2w+4+2w      (found by multiplying the 2 to the w and the 2 inside the parenthesis using the distributive property)

 

p=4w+4            (found by combining like terms)

 

40=4w+4          (p=40 was given in the question)

 

36=4w               (found by subtracting 4 from both sides of the equation)

 

9=w                  (found by dividing both sides of the equation by 4)

 

Step Three: Now we will plug in the value we got for w into formula 1

 

p=2l+2w     (formula 1)

 

40=2l+2w    (p=40 was given in the question)

 

40=2l+2(9)      (found by plugging the value found for w in step three in for w)

 

40=2l+18         (found by multiplying 2 and 9 together)

 

22=2l               (found by subtracting 18 from both sides of the equation)

 

11=l                 (found by dividing both sides of the equation by 2)

 

Solution: from substitution, we have found that w=9 and l=11

 

Check: To check your answer, plug all known values into formula 1

 

p=2l+2w  (formula 1)

 

40=2(11)+2(9)    (found by plugging in known values)

 

40=22+18           (found using multiplication)

 

40=40                  (found using addition)

 

 

Since both sides of your equation are equal, you know that the values you found for l and w are accurate, so your dimensions are 9ftx11ft

 

 

Answer 3

Perimeter is what we get when we add the length of all sides of an object. In the case of a rectangle, we will add 4 sides, 2 x length and 2 x width, together. 40 ft. is shown to be the perimeter of the rug, so now we need to figure out what the length and width are.

 

"The length is 2 ft. longer than the width." This portion of the question implies that the value of the length is dependent on the value of the width. Since the length is 2 ft. longer then the width, we will be adding 2 ft. to whatever value the width is in order to get the value of the length.

 

Let's use the variable W to express the width. Following this format, the width is now simply W and the length is now   W + 2.

 

BUT, we will need to double each of these values in order to get the total perimeter.

 

Therefore, W + W + 2 + W + W + 2 = 40 ft.

 

Combine like terms to get 4W + 4 = 40ft.

 

Subtract 4 from both sides to get 4W = 36ft.

 

Divide both sides by 4 to get W = 9

 

The question asked for the dimensions. The width = 9 and the length = W + 2 = 11

 

11 ft. x 9 ft.

 

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