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Chapter 8: Problem 51

What are the perimeter and area of a rectangle with length of \(53 \sqrt \text { centimeters and width of } 32 \sqrt \text { centimeters? }\)

Short Answer

Expert verified
Perimeter is \( 170 \sqrt{} \) cm; Area is \( 1696 \) units.

Step by step solution

01

Understand the problem

We need to find both the perimeter and area of a rectangle. We are given that the length is \( 53 \sqrt{} \) centimeters and the width is \( 32 \sqrt{} \) centimeters. However, it seems there's a missing component in the units. Assuming these are literal values, proceed with calculations under standard assumptions unless specified otherwise.
02

Perimeter Formula

The formula to find the perimeter of a rectangle is \( P = 2(l + w) \), where \( l \) is the length and \( w \) is the width of the rectangle.
03

Calculate Perimeter

Substitute the given length and width into the formula: \( P = 2(53 \sqrt{} + 32 \sqrt{}) \). Simplify the expression to find \( P \).
04

The Concept of Area

To find the area of a rectangle, use the formula \( A = l \times w \), where \( l \) is length and \( w \) is width.
05

Calculate Area

Substitute the given length and width into the area formula: \( A = 53 \sqrt{} \times 32 \sqrt{} \). Simplify the expression to find \( A \).
06

Finalize Calculations

After substituting and simplifying, the perimeter \( P \) and area \( A \) will give us the final results.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perimeter Calculation
When calculating the perimeter of a rectangle, it's important to know the formula: \( P = 2(l + w) \). The perimeter is the total distance around the rectangle. In our problem, we have a length \( l = 53 \sqrt{} \) centimeters and a width \( w = 32 \sqrt{} \) centimeters.
To find the perimeter, simply add the length and width together, then multiply by 2. This represents the sum of all sides:
  • Step 1: Add the length and width: \( 53 \sqrt{} + 32 \sqrt{} \)
  • Step 2: Multiply the result by 2: \( P = 2 \times (53 \sqrt{} + 32 \sqrt{}) \)
  • Step 3: Simplify the expression to get the final perimeter value.
Simplification in this case might involve resolving any root expressions, but if it's unclear, you can leave it in this form to convey an expression that awaits further specification.
Area Calculation
Calculating the area of a rectangle involves multiplying the length by the width. The area gives us the size of the surface within the rectangle's boundaries. For this rectangle, the formula comes in handy: \( A = l \times w \).
Our rectangle's given length is \( 53 \sqrt{} \) centimeters and the width is \( 32 \sqrt{} \) centimeters. Let's tackle this calculation step by step:
  • Step 1: Multiply the length and width directly: \( A = 53 \sqrt{} \times 32 \sqrt{} \)
  • Step 2: Calculate or simplify any root terms if possible for a clearer result.
This formula simplifies the multiplication of these expressions unless variables or constants clarify the form or units.
Algebraic Simplification
Algebraic simplification involves resolving expressions to their simplest form. In our context, with the given values \( 53 \sqrt{} \) and \( 32 \sqrt{} \), you might have to handle expressions involving square roots.
The goal of simplification is to make expressions more understandable and manageable:
  • If you encounter roots, try multiplying or dividing them as needed using rules of exponents and radicals.
  • Consider the expression of the form \((a\sqrt{b})\) to see if it can be further simplified.
  • Keep an eye on units or missing terms that could influence calculations which might require reevaluation.
In geometry problems, simplifying expressions allows for clearer answers and easier application of core formulas.

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