Absolutely! Let's work through the problem step by step to find the length of the rectangle, given that its perimeter is 76 and its length is three more than four times its width.
### Step 1: Define the Variables
Start by letting the width of the rectangle be [tex]\( w \)[/tex].
### Step 2: Express the Length in Terms of Width
According to the problem, the length [tex]\( l \)[/tex] of the rectangle is three more than four times its width. So, we can write this relationship as:
[tex]\[ l = 4w + 3 \][/tex]
### Step 3: Write the Formula for the Perimeter
The perimeter [tex]\( P \)[/tex] of a rectangle is given by the formula:
[tex]\[ P = 2l + 2w \][/tex]
### Step 4: Plug in the Given Perimeter
We know that the perimeter is 76. Therefore, we can substitute [tex]\( l \)[/tex] in the perimeter formula:
[tex]\[ 2l + 2w = 76 \][/tex]
### Step 5: Substitute the Expression for Length
Using the expression for [tex]\( l \)[/tex] from Step 2, we substitute [tex]\( 4w + 3 \)[/tex] for [tex]\( l \)[/tex] in the perimeter equation:
[tex]\[ 2(4w + 3) + 2w = 76 \][/tex]
### Step 6: Simplify the Equation
Distribute the 2 through the parentheses:
[tex]\[ 8w + 6 + 2w = 76 \][/tex]
Combine like terms:
[tex]\[ 10w + 6 = 76 \][/tex]
### Step 7: Solve for Width [tex]\( w \)[/tex]
Subtract 6 from both sides to isolate the terms with [tex]\( w \)[/tex]:
[tex]\[ 10w = 70 \][/tex]
Divide both sides by 10:
[tex]\[ w = 7 \][/tex]
So, the width of the rectangle is 7 units.
### Step 8: Find the Length
Now that we have the width, we can find the length using the expression [tex]\( l = 4w + 3 \)[/tex]:
[tex]\[ l = 4(7) + 3 \][/tex]
[tex]\[ l = 28 + 3 \][/tex]
[tex]\[ l = 31 \][/tex]
### Conclusion
Therefore, the length of the rectangle is:
[tex]\[ \boxed{31 \text{ units}} \][/tex]
