The length of a rectangle is three less than four times its width. If its perimeter is 54, then find its length and width
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Step 1: Assign variables:
Let's represent the width of the rectangle as "w".
Therefore, the length of the rectangle can be represented as "4w - 3".
Step 2: Define the perimeter of the rectangle:
The perimeter of a rectangle is calculated by adding all the sides. In this case, the formula for the perimeter is:
Perimeter = 2(length) + 2(width)
Substituting the values from step 1, we get:
54 = 2(4w - 3) + 2w
Step 3: Simplify and solve the equation:
Expanding the equation, we get:
54 = 8w - 6 + 2w
Combine like terms:
54 = 10w - 6
Add 6 to both sides:
60 = 10w
Divide both sides by 10:
w = 6
Step 4: Find the length:
Substitute the value of width (w = 6) into the equation we found in step 1:
Length = 4w - 3 = 4(6) - 3 = 24 - 3 = 21
Step 5: Conclusion:
The width of the rectangle is 6 units and the length is 21 units.
To find the length and width of the rectangle, we can set up equations based on the given information. Let's denote the width as "w" and the length as "l".
We are given that the length is three less than four times the width. Therefore, we can write the equation:
l = 4w - 3 (Equation 1)
We also know that the perimeter of a rectangle is given by the formula:
perimeter = 2(l + w)
Substituting the value of "l" from Equation 1, we get:
54 = 2((4w - 3) + w)
Now, let's solve this equation to find the value of "w" (width).
First, distribute the 2:
54 = 2(4w - 3 + w)
54 = 2(5w - 3)
Next, simplify:
54 = 10w - 6
Now, isolate the variable:
54 + 6 = 10w
60 = 10w
Finally, solve for "w":
w = 60 / 10 = 6
We have found the width of the rectangle, which is 6.
To find the length, substitute this value back into Equation 1:
l = 4w - 3
l = 4(6) - 3
l = 24 - 3
l = 21
Therefore, the length of the rectangle is 21.
In summary, the width of the rectangle is 6 units and the length is 21 units.
P = 2L + 2W
54 = 2(4W - 3) + 2W
54 = 10W - 6
60 = 10W
6 = W