George built a rectangular pen for his rabbit such that the length is 1 ft more than four times the width. If the perimeter is 292 ft, what are the dimensions of the pen?
2(w + 4w+1) = 292
To solve this problem, we can use the information given to set up a system of equations and then solve for the dimensions of the pen.
Let's start by assigning variables to the unknowns:
Let
- width of the pen be "W" in feet,
- length of the pen be "L" in feet.
We are given two pieces of information:
1. The length is 1 foot more than four times the width:
L = 4W + 1 (Equation 1)
2. The perimeter is 292 feet:
Perimeter = 2L + 2W = 292 (Equation 2)
Now, we have a system of equations with two unknowns (L and W).
We can solve this system of equations to find the values of L and W.
Step 1: Substitute the value of L from Equation 1 into Equation 2:
2(4W + 1) + 2W = 292
Simplifying the equation:
8W + 2 + 2W = 292
10W + 2 = 292
Step 2: Subtract 2 from both sides of the equation:
10W = 290
Step 3: Divide both sides by 10 to isolate W:
W = 29
Step 4: Substitute the value of W into Equation 1 to find L:
L = 4(29) + 1
L = 116 + 1
L = 117
The width (W) of the pen is 29 feet, and the length (L) of the pen is 117 feet.
Let's solve this step-by-step:
Step 1: Let's represent the width of the pen as "w" ft.
Step 2: According to the given information, the length of the pen is 1 ft more than four times the width. So, the length can be represented as "4w + 1".
Step 3: The formula for the perimeter of a rectangle is: Perimeter = 2(length + width)
Step 4: Substituting the values into the formula, we can write the equation: 292 = 2(4w + 1 + w)
Step 5: Simplifying the equation, we get: 292 = 2(5w + 1)
Step 6: Distributing the 2, we get: 292 = 10w + 2
Step 7: Subtracting 2 from both sides of the equation, we get: 290 = 10w
Step 8: Dividing both sides of the equation by 10, we get: w = 29
Step 9: Now that we have the value of the width, we can substitute it back into the equation for the length to find the length. Using the expression for the length, we have: Length = 4w + 1 = 4(29) + 1 = 116 + 1 = 117.
Therefore, the dimensions of the rectangular pen are:
Width = 29 ft
Length = 117 ft