The perimeter of the rectangle below is 150 units. Find the length of side \(\overline{QR}\). The sides of the rectangle are labeled as follows: - One side: \(3x + 3\) - Adjacent side: \(4x + 2\) Find \(\overline{QR}\) without variables.
Answer (1)
To find the length of side QR of the rectangle, we need to understand how to use the information given about the perimeter.
The perimeter P of a rectangle is calculated using the formula:
P = 2 × ( length + width )
We are told the perimeter is 150 units, and the sides are labeled as follows:
One side: 3 x + 3
Adjacent side: 4 x + 2
This means the length and width of the rectangle can be thought of as 3 x + 3 and 4 x + 2 respectively.
Substituting into the perimeter formula, we get:
150 = 2 × (( 3 x + 3 ) + ( 4 x + 2 ))
First, simplify the expression inside the parentheses:
( 3 x + 3 ) + ( 4 x + 2 ) = 3 x + 3 + 4 x + 2 = 7 x + 5
So, the equation becomes:
150 = 2 × ( 7 x + 5 )
Next, distribute the 2:
150 = 14 x + 10
Subtract 10 from both sides to isolate the term with x :
140 = 14 x
Divide both sides by 14 to solve for x :
x = 14 140 = 10
Now, substitute x = 10 back into one of the expressions for the side length, 3 x + 3 :
3 x + 3 = 3 ( 10 ) + 3 = 30 + 3 = 33
Thus, the length of side QR is 33 units.
