The length of a rectangle is 10 more than three times its width. The perimeter of the rectangle is 108. Find the dimensions of the rectangle.
Answer (2)
Define the width as w and express the length as l = 3 w + 10 .
Use the perimeter formula P = 2 l + 2 w and substitute the given perimeter P = 108 .
Solve the equation 108 = 2 ( 3 w + 10 ) + 2 w for w , which gives w = 11 .
Substitute w = 11 into l = 3 w + 10 to find l = 43 . The dimensions are w = 11 , l = 43 .
Explanation
Problem Analysis Let's break down this problem step by step to find the dimensions of the rectangle.
Define Variables Let's define our variables:
Let w represent the width of the rectangle.
Since the length is 10 more than three times the width, we can express the length l as:
l = 3 w + 10
Perimeter Formula The formula for the perimeter P of a rectangle is:
P = 2 l + 2 w
Substitute Length Now, substitute the expression for l into the perimeter formula:
P = 2 ( 3 w + 10 ) + 2 w
Substitute Perimeter We are given that the perimeter P = 108 . Substitute this value into the equation:
108 = 2 ( 3 w + 10 ) + 2 w
Solve for Width Now, let's solve for w :
108 = 6 w + 20 + 2 w
108 = 8 w + 20
Subtract 20 from both sides:
88 = 8 w
Divide by 8:
w = 11
Solve for Length Now that we have the width, we can find the length:
l = 3 w + 10
l = 3 ( 11 ) + 10
l = 33 + 10
l = 43
Final Dimensions So, the dimensions of the rectangle are:
Width: w = 11
Length: l = 43
Examples
Understanding the dimensions of rectangles is crucial in many real-world applications. For example, architects use these calculations when designing buildings and rooms, ensuring that spaces are functional and aesthetically pleasing. Similarly, landscapers use these principles to plan gardens and outdoor areas, optimizing space and resource allocation. Even in everyday tasks like framing a picture or building a shelf, knowing how to calculate dimensions helps ensure accuracy and efficiency.
The dimensions of the rectangle are 11 units for the width and 43 units for the length, found by defining variables, substituting into the perimeter formula, and solving the resulting equations.
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