Solving for the Dimensions of a Rectangle Given Perimeter and Proportional Relationship
When faced with a problem that involves finding the dimensions of a rectangle given its perimeter and a proportional relationship between its length and width, it's important to use both the perimeter formula and the given proportion to solve the problem. In this article, we will solve a specific problem and explore the steps involved.
Problem Statement
The perimeter of a rectangle is 11 cm, and the length is 3 cm more than half the width. What are the dimensions of the rectangle?
Step-by-Step Solution
To find the dimensions of the rectangle, we can set up equations based on the information provided. Let the width be (w) cm.
Then the length (l) can be expressed as:
(l frac{w}{2} 3)
The formula for the perimeter (P) of a rectangle is:
(P 2l 2w)
Given that the perimeter is 11 cm, we have:
(2l 2w 11)
Substituting the expression for (l) into the perimeter formula:
(2left(frac{w}{2} 3right) 2w 11)
Simplifying this equation:
(2 cdot frac{w}{2} 6 2w 11)
(w 6 2w 11)
(3w 6 11)
Now, solve for (w):
(3w 11 - 6)
(3w 5)
(w frac{5}{3} approx 1.67 , text{cm})
Substitute (w) back to find (l):
(l frac{frac{5}{3}}{2} 3 frac{5}{6} 3 frac{5}{6} frac{18}{6} frac{23}{6} approx 3.83 , text{cm})
Thus, the dimensions of the rectangle are:
Width: (frac{5}{3}) cm, approximately 1.67 cm. Length: (frac{23}{6}) cm, approximately 3.83 cm.In conclusion, the dimensions of the rectangle are approximately 1.67 cm in width and 3.83 cm in length.
Alternative Solutions
Another way to approach the problem involves solving a system of equations directly:
Let (a) be the width, and (b) be the length.
(2a b 11)
(b frac{a}{2} 3)
Substituting (b) into the first equation:
[2a left(frac{a}{2} 3right) 11]
[2a frac{a}{2} 8]
[frac{5a}{2} 8]
[a frac{16}{5} 3.2]
[b frac{3.2}{2} 3 1.6 3 4.6]
These dimensions do not match our earlier solution, indicating a potential different interpretation of the problem statement.
Conclusion
The solution we derived in the first part by directly using the perimeter and the given proportional relationship is correct. If the problem interpretation is strictly as stated, then the correct dimensions of the rectangle are approximately 1.67 cm (width) and 3.83 cm (length).