What is the measure of the length of a rectangle with a width of 3x + 4y + 1 and a perimeter of 24x + 18y + 32?
p = 2*width + 2*length
24x + 18y + 32 = 2(3x + 4y + 1) + 2*length
24x + 18y + 32 = 6x + 8y + 2 + 2*length
18x + 10y + 30 = 2*length
9x + 5y + 15 = length
To find the length of the rectangle, we need to use the perimeter formula, which states that the perimeter of a rectangle is equal to twice the sum of its length and width.
Let's start by assigning variables to the length and width. Let L represent the length and W represent the width.
Given:
Width (W) = 3x + 4y + 1
Perimeter = 24x + 18y + 32
The perimeter formula can be written as:
2(L + W) = Perimeter
Substituting the values, we have:
2(L + (3x + 4y + 1)) = 24x + 18y + 32
Now, let's simplify the equation:
2(L + 3x + 4y + 1) = 24x + 18y + 32
2L + 6x + 8y + 2 = 24x + 18y + 32
2L + 6x + 8y = 24x + 18y + 30
Moving all the variables to one side, we get:
2L = 24x + 18y + 30 - 6x - 8y
2L = 18x + 10y + 30
Dividing both sides by 2, we find:
L = (18x + 10y + 30) / 2
Therefore, the measure of the length of the rectangle with a width of 3x + 4y + 1 and a perimeter of 24x + 18y + 32 is (18x + 10y + 30) / 2.