The perimeter of a patio is to be 96 feet. Determine the dimensions of the patio if the length is to be 6 feet less than twice the width

Answer 1

Hi Sabrina!

This problem requires two formulas. The first can be found by the sentence "The perimeter of a patio is to be 96 feet". The perimeter of the patio is the total length of all of the sides added up, so the following formula can be created to represent the perimeter of the patio:

96 feet = (2 * Length) + (2 * Width)

The second sentence tells us that the length of the patio is to be "6 feet less than twice the width". In other words, if we take the width of one side of the patio and multiply it by 2, this number will be 6 feet greater than the length of one side of the patio. Therefore, we can set up a formula to represent the length of the patio as follows:

Length = (2 * Width) - 6 feet

Now that we know how the length and the width of the patio are correlated, we can input our second formula into our first formula (replace "Length" with the formula for Length):

96 feet = (2 * [(2 * Width) - 6 feet]) + (2 * Width)

Now that we have only one variable (Width), we can solve for it as follows:

1. Simplify the right side of the equation

96 feet = (2 * [(2 * Width) - 6 feet]) + (2 * Width)

96 feet = (4 * Width) - 12 feet + (2 * Width)

96 feet = (6 * Width) - 12 feet

1. Get our variable "Width" on its own on the right side of the equation

96 feet = (6 * Width) - 12 feet

96 feet + 12 feet = (6 * Width)

108 feet = (6 * Width)

108 feet / 6 = Width

18 feet = Width

Now we know the width of one side of the patio: 18 feet. We also know that the Length of one side of the patio is 6 less than twice the width, so we can input our width into the formula we created before to find Length:

Length = (2 * Width) - 6 feet

Length = (2 * 18 feet) - 6 feet

Length = 36 feet - 6 feet

Length = 30 feet

Now we know what the length and width of the patio should be (30 and 18, respectively). We can check this figure against the original statement that the patio should have a perimeter of 96 feet as follows:

96 feet = (2 * Length) + (2 * Width)

96 feet = (2 * 30 feet) + (2 * 18 feet)

96 feet = 60 feet + 36 feet

This kind of problem shows how a formula with multiple variables can be solved once you know the correlation between the variables. Since we knew how the length was related to the width, we could substitute length in the main formula with a formula that refers to width. Which allowed us to solve for width and then the length.

Hope that helps!

Answer 2

Hello Sabrina,

Start solving the problem by using the formula for perimeter of a rectangle. Perimeter equals 2 times the length plus 2 times the width.

P = 2L + 2W

You are given P = 96, but no numbers for length and width. However, you are given a relationship between the two and so you need to pick appropriate variables.

I would make the width W (or x or whatever variable you would like) since the length depends on the width. The problem tells you that the length is 6 feet less, so you will have a -6 in there. But first you must multiply the width by 2 since it is 6 less than twice the width.

Length (or L) = twice the width less 6

OR

L = 2W - 6

Now you can substitute 96 in for P and 2W - 6 in for L in the original equation to solve.

P = 2L + 2W

96 = 2(2W -6) + 2W

96 = 4W - 12 + 2W

96 = 6W - 12

96 + 12 = 6W

108 = 6W

W = 18

Always double check your work by putting the answer back into the equation to make sure it works....in this case it does!

96 = 2(2(18) -6) + 2(18)

96 = 2(36 -6) + 36

96 = 2(30) +36

96 = 60 + 36

96 = 96

Hope this helps!

Answer 3

Let x = the width

2x-6 = the length (based on the conditions given)

Perimeter = width + width + length + length

96 = x+x+2x-6+2x-6 Combine like terms

96=6x-12 add 12 to both sides

108=6x Divide both sides by 6

x = 18 ft, this is the width

Now we can also find the length

2x-6 = 2(18)-6 = 30 ft, this is the length

Answer 4

Perimeter = 96 = 2(L+W), L=2W-6,

L=length, W= width, substitute 2W-6 for L

96=2(2W-6+W) = 6W-12

6W = 108

W = 108/6 = 18 feet

L=2W-6 = 2(18)-6 = 36-6 = 30 feet

check the answers

96=2(30+18)= 2(48) = 96

L=2W-6, 30=2(18)-6 = 36-6=30