Mr. Reed is drawing a blueprint of a rectangular patio. The width of the patio is
feet shorter than twice its length. The perimeter of the patio is
feet. What is the length of the patio?
The Correct Answer and Explanation is:
To determine the length of the patio, let’s define variables for the dimensions:
Let LL be the length of the patio and WW be the width. The problem states that the width is 40 3/4 feet less than twice the length:
W=2L−4034W = 2L – 40 \frac{3}{4}
The perimeter of a rectangle is calculated using the formula:
P=2L+2WP = 2L + 2W
Substituting the given perimeter value of 86 1/2 feet:
2L+2W=86122L + 2W = 86 \frac{1}{2}
Next, replace WW with its expression:
2L+2(2L−4034)=86122L + 2(2L – 40 \frac{3}{4}) = 86 \frac{1}{2}
Expanding:
2L+4L−8112=86122L + 4L – 81 \frac{1}{2} = 86 \frac{1}{2}
Adding 81 1/2 to both sides:
6L=1686L = 168
Dividing both sides by 6:
L=28L = 28
Thus, the length of the patio is 28 feet.
Explanation:
This problem involves understanding the relationship between the length and width of a rectangle and applying algebraic principles to solve for the unknown dimension. The width is expressed in terms of the length, and this expression is substituted into the perimeter equation. By expanding, simplifying, and isolating the variable, we arrive at the correct answer.
Using a systematic approach ensures accuracy in solving such problems. This method aligns well with real-world applications, such as architectural design and construction planning, where precise measurements are crucial. The equation-based solution demonstrates how algebra and geometry intersect in practical scenarios, reinforcing the importance of structured problem-solving techniques.
