The perimeter of a rectangle is 76 feet. If the width is doubled and the length is halved, the new rectangle will have a perimeter of 62 feet. Find the dimensions of the original rectangle. Define your variables.
old rectangle:
length : y
width : x
so 2x + 2y = 76 or
x+y = 38
new rectangle:
2x by y/2
2(2x) + 2(y/2) = 62
4x + y = 62
subtract them:
3x = 24
x = 8 ,then y = 30
Let's assume that the length of the original rectangle is L feet and the width is W feet.
According to the problem, the perimeter of the original rectangle is 76 feet. The formula for the perimeter of a rectangle is given by:
Perimeter = 2 * (Length + Width)
So, we can write the equation as:
76 = 2L + 2W ---(1)
Now, let's consider the new rectangle. It is given that the width is doubled and the length is halved. So, the new width is 2W feet and the new length is (1/2)L feet.
The perimeter of the new rectangle is 62 feet. Using the same formula for the perimeter, we can write the equation as:
62 = 2 * ((1/2)L + 2W) ---(2)
Now, we have a system of two equations (equations 1 and 2) with two variables (L and W).
Let's solve the system of equations to find the values of L and W.
From equation 1, we can express W in terms of L:
W = (76 - 2L) / 2
Substituting this value of W into equation 2, we get:
62 = 2 * ((1/2)L + 2((76 - 2L) / 2))
Simplifying,
62 = L + 76 - 2L
Combine like terms,
62 = 76 - L
Rearranging,
L = 76 - 62
L = 14
Substituting this value of L into equation 1, we can find W:
76 = 2(14) + 2W
76 = 28 + 2W
48 = 2W
W = 24
Therefore, the dimensions of the original rectangle are:
Length (L) = 14 feet
Width (W) = 24 feet
To solve this problem, let's define the variables:
Let's say the original length of the rectangle is L feet.
Let's say the original width of the rectangle is W feet.
According to the problem, the perimeter of the original rectangle is 76 feet, so we can write the equation:
2L + 2W = 76 ---- Equation (1)
Next, the problem states that if the width is doubled (2W) and the length is halved (L/2), the new rectangle will have a perimeter of 62 feet. So we can write the equation for the new rectangle:
2(L/2) + 2(2W) = 62
Simplifying this equation, we get:
L + 4W = 62 ---- Equation (2)
Now we have a system of two equations (Equations 1 and 2) with two unknowns (L and W), so let's solve it.
We can start by solving Equation (2) for L:
L = 62 - 4W
Now substitute this value of L into Equation (1):
2(62 - 4W) + 2W = 76
Expanding and simplifying this equation:
124 - 8W + 2W = 76
124 - 6W = 76
Subtracting 124 from both sides:
-6W = -48
Dividing both sides by -6:
W = 8
Now substitute the value of W = 8 into Equation (1) to find L:
2L + 2(8) = 76
2L + 16 = 76
Subtracting 16 from both sides:
2L = 60
Dividing both sides by 2:
L = 30
Therefore, the original rectangle has dimensions of length = 30 feet and width = 8 feet.