Problem 75 - Calculus: One and Several Variables

Question

A string 28 inches long is to be cut into two pieces. one piece to form a square and the other to form a circle. Express the total area enclosed by the square and circle as a function of the perimeter of the square.

Step-by-Step Solution

Answer

The total area enclosed by the square and circle as a function of the perimeter of the square is \( A = (P^2 / 16) + \pi ((28 - P) / (2\pi))^2 \).

Step 1: Express square's side from its perimeter

Given that the square's perimeter is denoted by \( P \), then the side length of the square \( S \) would be \( P/4 \), because a square has four equal sides.

Step 2: Express circle's radius from its circumference

Knowing that the total string length is 28 inches and the square's perimeter is \( P \), then the circumference of the circle is \( 28 - P \). As the length of a circle is expressed by the formula \( C = 2\pi r \), we can solve for \( r \) to find the radius of the circle: \( r = (28 - P) / 2\pi \).

Step 3: Calculate the square's area

Taking the side length of the square from step 1, we can calculate the square's area \( A_S \) using the formula \( A_S = S^2 \). When substituting \( S = P/4 \), we find the square's area: \( A_S = (P/4)^2 \).

Step 4: Calculate the area of the circle

Using the derived radius from step 2, compute the area of the circle \( A_C \) with the following formula: \( A_C = \pi r^2 \). Substituting \( r = (28 - P) / (2\pi) \), we obtain the area of the circle: \( A_C = \pi ((28 - P) / (2\pi))^2 \).

Step 5: Calculate the total area enclosed by the square and circle as a function of the perimeter of the square

The total area of the shapes, denoted as \( A \), is simply the sum of the square's area \( A_S \) and the circle's area \( A_C \). Therefore: \( A = A_S + A_C \) which equates to \( A = (P^2 / 16) + \pi ((28 - P) / (2\pi))^2 \).

Key Concepts

Geometric Shapes

Understanding geometric shapes is crucial in solving problems related to area and perimeter. In geometry, shapes such as squares and circles are defined by specific properties.
For a square:

• It has four equal sides.
• All the angles are right angles.
• The perimeter is calculated by adding the lengths of all sides.

For a circle, on the other hand:

• The distance around the circle is called the circumference.
• It is defined by a radius, which is the distance from the center to any point on the circle.
• The relationship between the circumference and the radius is through the constant \( \pi \), a key concept in circular shapes.

Recognizing these basics helps us manipulate them mathematically to solve more complex problems involving the combined areas of these shapes.

Perimeter and Area

Perimeter and area are fundamental concepts in geometry. The perimeter of a shape is the total length around it. For example, the perimeter \( P \) of a square can be determined if you know the side length \( S \), by the formula \( P = 4S \). Conversely, if you have the perimeter, the side length can be found by \( S = P/4 \).

Likewise, knowing the perimeter of a part of a string budgeted to form a circle gives the circumference of that circle, allowing us to find the radius \( r \) with the formula \( C = 2\pi r \). This relationship is instrumental in managing situations where the total available length must be divided between different shapes.

The area of these shapes is the space they occupy. For a square, the area \( A_S \) is found by squaring the side length: \( A_S = S^2 \). For a circle, the area \( A_C \) is found using the formula \( A_C = \pi r^2 \). Understanding these relationships is essential for calculating areas precisely, especially when the resources are shared.

Functions of a Variable

In optimization problems like this one, we deal with functions of a variable to express relationships between different quantities. Here, we aim to express the total area of the square and circle as a function of the perimeter of the square. This involves defining a function \( A(P) \) where \( P \) is the variable representing the perimeter of a square.

We derive side length of the square \( S \) and radius \( r \) of the circle based on \( P \), and then use those to express the areas. The formula to find the total area \( A \) becomes a combination of the squares' and circles' areas:

• Square's area \( (P/4)^2 \).
• Circle's area \( \pi ((28 - P) / (2\pi))^2 \).

So, \( A(P) = (P^2 / 16) + \pi ((28 - P) / (2\pi))^2 \). Creating such functions allows us to pinpoint the relationship and see how altering \( P \) impacts the total area, which is vital in optimization.