The perimeter of a rectangle is $72 \sqrt{5} mm$ and the width of the rectangle is $3 \sqrt{20} mm$. Rhonda thinks the length is $36 \sqrt{5}-3 \sqrt{20} mm$ and Konrad thinks the length is $30 \sqrt{5} mm$. Who is correct? Explain fully.
Answer (1)
Simplify the width: w = 3 20 = 6 5 .
Apply the perimeter formula: 72 5 = 2 l + 12 5 .
Solve for the length: l = 30 5 .
Both Rhonda's simplified length and Konrad's length match the calculated length: l = 30 5 .
Explanation
Problem Analysis Let's analyze the problem. We are given the perimeter of a rectangle and its width. We need to determine who, Rhonda or Konrad, correctly calculated the length of the rectangle. We will use the formula for the perimeter of a rectangle, which is P = 2 l + 2 w , where P is the perimeter, l is the length, and w is the width.
Given Information We are given: Perimeter P = 72 5 mm Width w = 3 20 mm
We need to find the length l .
Simplify the Width First, let's simplify the expression for the width: w = 3 20 = 3 4 × 5 = 3 × 2 5 = 6 5 mm
Apply Perimeter Formula Now, let's use the perimeter formula and solve for the length: P = 2 l + 2 w 72 5 = 2 l + 2 ( 6 5 ) 72 5 = 2 l + 12 5
Isolate the Length Term Subtract 12 5 from both sides: 2 l = 72 5 − 12 5 2 l = 60 5
Solve for Length Divide both sides by 2: l = 2 60 5 l = 30 5 mm
Compare with Proposed Lengths Now, let's compare our result with Rhonda's and Konrad's proposed lengths: Rhonda's proposed length: 36 5 − 3 20 = 36 5 − 3 ( 2 5 ) = 36 5 − 6 5 = 30 5 mm Konrad's proposed length: 30 5 mm
Both Rhonda's and Konrad's proposed lengths are equal to the calculated length.
Conclusion Since both Rhonda's simplified length and Konrad's length match the calculated length, both are technically correct. However, Konrad's answer is already simplified, while Rhonda's requires simplification to arrive at the correct answer.
Examples
Understanding perimeters and lengths of rectangles is crucial in many real-world applications. For example, when designing a rectangular garden, you need to calculate the amount of fencing required (perimeter) based on the desired length and width. Similarly, in construction, calculating the dimensions of a room or a building often involves using the perimeter formula to ensure accurate measurements and material usage. This concept is also applicable in interior design when arranging furniture or determining the size of rugs needed to fit a specific space.
