Which of the following is a simplified expression of perimeter for this shape:
[tex]\frac{5 \sqrt{2}-1}{\sqrt{5}+3 \sqrt{2}}[/tex]
The length is : [tex]$\sqrt{5}+3 \sqrt{2}$[/tex]
The width is: [tex]$5 \sqrt{2}-1$[/tex]
Answer (1)
Calculate the perimeter using the formula: P = 2 ∗ ( l e n g t h + w i d t h ) .
Substitute the given length 5 + 3 2 and width 5 2 − 1 into the formula: P = 2 ∗ (( 5 + 3 2 ) + ( 5 2 − 1 )) .
Simplify the expression: P = 2 5 + 16 2 − 2 .
The simplified expression for the perimeter is: 16 2 + 2 5 − 2 .
Explanation
Understanding the Problem We are given the length and width of a shape and asked to find the simplified expression for its perimeter. The length is 5 + 3 2 and the width is 5 2 − 1 . The perimeter of a shape is given by the formula P = 2 ∗ ( l e n g t h + w i d t h ) . We need to substitute the given length and width into the formula and simplify the expression.
Substituting Values Substitute the given length and width into the perimeter formula: P = 2 ∗ (( 5 + 3 2 ) + ( 5 2 − 1 ))
Simplifying the Expression Simplify the expression inside the parenthesis by combining like terms: P = 2 ∗ ( 5 + 3 2 + 5 2 − 1 ) P = 2 ∗ ( 5 + 8 2 − 1 )
Distributing the Constant Distribute the 2 to each term inside the parenthesis: P = 2 5 + 16 2 − 2
Comparing with Options Now, we compare the simplified perimeter expression 2 5 + 16 2 − 2 with the given options. Option 1: 8 2 + 5 − 2 Option 2: 5 10 − 5 − 3 2 + 30 We can see that our calculated perimeter 2 5 + 16 2 − 2 is not exactly the same as either of the given options. However, let's examine the first option more closely. It seems there was an error in the provided options. The correct simplified expression should be 16 2 + 2 5 − 2 . However, if we look at the first option, we can factor out 2 and rewrite the expression as: 2 ( 8 2 + 5 − 1 ) . This is not the same as our expression.
Let's approximate the value of our expression: 2 5 + 16 2 − 2 ≈ 2 ( 2.236 ) + 16 ( 1.414 ) − 2 ≈ 4.472 + 22.624 − 2 ≈ 25.096
Now, let's approximate the values of the given options: Option 1: 8 2 + 5 − 2 ≈ 8 ( 1.414 ) + 2.236 − 2 ≈ 11.312 + 2.236 − 2 ≈ 11.548 Option 2: 5 10 − 5 − 3 2 + 30 ≈ 5 ( 3.162 ) − 2.236 − 3 ( 1.414 ) + 30 ≈ 15.81 − 2.236 − 4.242 + 30 ≈ 39.332
None of the options match the simplified expression we derived. However, if the length was 5 + 8 instead of 5 + 3 2 , then the perimeter would be: P = 2 (( 5 + 8 ) + ( 5 2 − 1 )) = 2 ( 5 + 2 2 + 5 2 − 1 ) = 2 ( 5 + 7 2 − 1 ) = 2 5 + 14 2 − 2
If the width was 5 − 1 instead of 5 2 − 1 , then the perimeter would be: P = 2 (( 5 + 3 2 ) + ( 5 − 1 )) = 2 ( 2 5 + 3 2 − 1 ) = 4 5 + 6 2 − 2
Given the options, it seems there might be a typo in the problem statement. Assuming the first option 8 2 + 5 − 2 is the closest to the correct answer, we can assume that the length and width were different. However, based on the given information, the correct simplified expression for the perimeter is 16 2 + 2 5 − 2 .
Examples
Perimeter calculations are essential in various real-world applications, such as fencing a garden, framing a picture, or determining the amount of material needed to create a border around a room. Understanding how to calculate and simplify perimeter expressions allows us to efficiently plan and execute these tasks, saving time and resources. For instance, if you're building a rectangular garden with dimensions 5 + 3 2 meters and 5 2 − 1 meters, calculating the perimeter helps you determine the exact length of fencing required, ensuring you purchase the right amount of materials.
