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Chapter 3: Problem 13

Minimum Perimeter, find the length and width of a rectangle that has the given area and a minimum perimeter. Area: 49 square feet

Short Answer

Expert verified
The length and width of the rectangle, which is a square in this case, would both be 7 feet.

Step by step solution

01

Understand the problem

The problem is asking to find the length and width of a rectangle that has an area of 49 square feet and the smallest possible perimeter. We can infer from the properties of shapes that the rectangle would take the form of a square, as squares have the smallest possible perimeter for a given area among all rectangles.
02

Identify the properties of a square

A square is a special type of rectangle where all sides are equal. Therefore, both its length and its width are the same, and they can be found by taking the square root of the area. In this case, the area of the square is 49 square feet. Consequently, the formula to find the length (l) and width (w) will be: l = w = sqrt(Area).
03

Calculate the length and width

By using the formula identified in step 2, plug in the given area of 49 square feet into the equation. So both the length and width would be calculated as follows: l = w = sqrt(49) = 7.

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Key Concepts

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

Rectangle Area
When thinking about a rectangle, a fundamental concept to understand is how to calculate its area. The area is determined by multiplying its length by its width. Mathematically, this can be expressed as:
  • \( \text{Area} = \text{Length} \times \text{Width} \)
For example, if a rectangle has a length of 10 feet and a width of 5 feet, then the area is:
  • \( 10 \times 5 = 50 \) square feet
In problems aimed at optimizing either the area or the perimeter, knowing this basic formula allows us to manipulate one variable if the other is fixed. Understanding the area is crucial to resolving challenges involving the dimensions of rectangles.
Square Properties
A square is a very special type of rectangle. All four sides are equal in length, making geometry problems sometimes easier when they involve squares. If a square has an area, the length of each side can be found by taking the square root of the area. This is because:
  • \( \text{Area of a square} = \text{Side length}^2 \)
  • \( \text{Side length} = \sqrt{\text{Area}} \)
Consider a square with an area of 49 square feet. The side length would be:
  • \( \sqrt{49} = 7 \) feet
The square's properties are especially useful in perimeter and area optimization problems, as squares inherently have the minimal perimeter for any given area compared to rectangles with different side lengths.
Perimeter Calculation
Calculating the perimeter of geometrical shapes is straightforward once you have the dimensions. For rectangles, including squares, the formula to find the perimeter is:
  • \( \text{Perimeter of a rectangle} = 2 \times (\text{Length} + \text{Width}) \)
In the case of a square, where length equals width, the formula simplifies to:
  • \( \text{Perimeter of a square} = 4 \times \text{Side length} \)
Using this formula, if a square has a side length of 7 feet, the perimeter would be:
  • \( 4 \times 7 = 28 \) feet
Knowing how to calculate the perimeter helps optimize shapes, like determining the smallest possible perimeter for a designated area, as in our exercise.
Geometry Optimization
Optimization in geometry often involves finding the most efficient way to use space or materials. In our problem, we wanted to achieve the smallest possible perimeter for a given area. The solution was a square with area 49 square feet, resulting in dimensions where both the length and width are 7 feet.
  • Why does a square provide the minimum perimeter? Because among rectangles with the same area, a square has greater side equality, minimizing the sum of the sides.
  • So for area-based perimeter optimization, converting to a square is very effective.
These principles are not just theoretical; they reduce material costs and provide optimal use of space, often used in fields like architecture and engineering for practical applications.

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Most popular questions from this chapter

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