The length of a rectangle is 8 meters less than 5 times its width. if the perimeter of the rectangle is at most 104 meters, find the greatest possible width of the rectangle.
P = 2L + 2W
104 = 2(5W - 8) + 2W
104 = 10W - 16 + 2W
120 = 12W
? = W
Let's break down the problem step-by-step:
Step 1: Assign variables
Let's assign variables to the width and length of the rectangle. Let w be the width (in meters) of the rectangle and L be the length (in meters) of the rectangle.
Step 2: Express the length in terms of the width
From the given information, we know that the length of the rectangle is 8 meters less than 5 times its width. Therefore, we can express the length in terms of the width as follows: L = 5w - 8.
Step 3: Express the perimeter in terms of the width
The perimeter of a rectangle is given by the formula P = 2w + 2L. Substituting the expression for the length from Step 2, we have P = 2w + 2(5w - 8).
Step 4: Simplify the expression for the perimeter
Expanding the expression, we get P = 2w + 10w - 16.
Step 5: Set up an inequality for the perimeter and solve for the width
We are given that the perimeter of the rectangle is at most 104 meters. Therefore, we can set up the following inequality: 2w + 10w - 16 ≤ 104.
Step 6: Solve the inequality
Combining like terms, we have 12w - 16 ≤ 104.
Adding 16 to both sides, we get 12w ≤ 120.
Dividing both sides by 12, we get w ≤ 10.
Therefore, the greatest possible width of the rectangle is 10 meters.
To solve this problem, we'll need to set up an equation using the given information.
Let's start by assigning variables to the width and length of the rectangle. Let's say the width is 'w' meters and the length is 'l' meters.
According to the problem, the length of the rectangle is 8 meters less than 5 times its width. This can be expressed as:
l = 5w - 8
The perimeter of a rectangle can be calculated by adding up the lengths of all four sides. In this case, the perimeter is given to be at most 104 meters. Since the length and width are both positive values, we can write the equation as:
2l + 2w ≤ 104
Now, substitute the value of 'l' from the first equation into the second equation:
2(5w - 8) + 2w ≤ 104
Simplify and solve the inequality:
10w - 16 + 2w ≤ 104
12w - 16 ≤ 104
12w ≤ 120
Divide both sides by 12:
w ≤ 10
Therefore, the greatest possible width of the rectangle is 10 meters.