Problem 69 - Algebra for College Students

Question

. The equation \(A(r)=\pi r^{2}\) expresses the area of a circular region as a function of the length of a radius \((r)\). Compute \(A(2), A(3), A(12)\), and \(A(17)\) and express your answers to the nearest hundredth. \(12.57 ; 28.27 ; 452.39 ; 907.92\)

Step-by-Step Solution

Answer

The areas are 12.57, 28.27, 452.39, and 907.92.

Step 1: Understanding the formula

The given formula \(A(r) = \pi r^2\) calculates the area of a circle based on its radius \(r\). \(\pi\) is a constant approximately equal to 3.14159. To find the area for specific radii, substitute these values into \(r\).

Step 2: Calculate A(2)

Substitute \(r = 2\) into the formula: \(A(2) = \pi \times 2^2 = \pi \times 4\). Using \(\pi \approx 3.14159\), calculate as follows: \(A(2) = 3.14159 \times 4 = 12.56636\). Round to the nearest hundredth to get \(12.57\).

Step 3: Calculate A(3)

Substitute \(r = 3\) into the formula: \(A(3) = \pi \times 3^2 = \pi \times 9\). Using \(\pi \approx 3.14159\), calculate as follows: \(A(3) = 3.14159 \times 9 = 28.27431\). Round to the nearest hundredth to get \(28.27\).

Step 4: Calculate A(12)

Substitute \(r = 12\) into the formula: \(A(12) = \pi \times 12^2 = \pi \times 144\). Using \(\pi \approx 3.14159\), calculate as follows: \(A(12) = 3.14159 \times 144 = 452.38896\). Round to the nearest hundredth to get \(452.39\).

Step 5: Calculate A(17)

Substitute \(r = 17\) into the formula: \(A(17) = \pi \times 17^2 = \pi \times 289\). Using \(\pi \approx 3.14159\), calculate as follows: \(A(17) = 3.14159 \times 289 = 907.91961\). Round to the nearest hundredth to get \(907.92\).

Key Concepts

Understanding Mathematical Computation

Mathematical computation involves performing calculations based on mathematical operations and formulas. To compute means to use arithmetic or algorithms to figure out a specific numerical result. In our exercise, we are computing the area of a circle given its radius.
The area formula is applied by using precise arithmetic: this involves multiplication, powers, and approximating constants like \( \pi \).
By focusing correctly on arithmetic order and simplifications, you can achieve accurate results.

• Start with powers: Calculate the square of the radius, \( r^2 \).
• Follow with multiplication: Multiply this by \( \pi \) (approximately 3.14159).
• Apply rounding: Adopt rounding as necessary to meet precision needs, like rounding to the nearest hundredth.

Exploring the Substitution Method

The substitution method is a straightforward technique used to find solutions for variables by replacing them with specific values. In mathematical computation, this method involves inserting certain numbers into given formulas or equations.
In this exercise, radius values are substituted into the circle area formula \( A(r) = \pi r^2 \). Let's look at how substitution helps in determining different circle areas:

• Choose the value for \( r \), the radius.
• Substitute \( r \) into the formula: In \( A(2) = \pi \times 2^2 \), \( r = 2 \).
• Calculate the expression: Perform mathematical operations like squaring and multiplying.
• Repeat for each new radius value: This allows for finding areas of different circles.

Substitution helps in actively engaging with equations, enabling steps towards concrete numerical results.

Rounding Numbers Effectively

Rounding numbers is a method used to simplify figures, making them easier to communicate and use. For this exercise, answers are rounded to the nearest hundredth to ensure clarity and consistency.
Here’s how rounding typically works: Identify the digit at the desired place value, look at the digit to its right, and decide if it remains the same or increases. For instance,

• If the digit to the right is 5 or greater, increase the target digit by 1, e.g., 12.56636 becomes 12.57.
• If less than 5, leave the target digit as is, e.g., 28.27431 becomes 28.27.

This operation is pivotal in mathematical computation to ascertain readability and accuracy of results. It's a fine balance between precision and simplicity.

The Radius of a Circle Uncovered

The radius of a circle is the key measurement from its center to any point on its circumference. Knowing this, the radius is fundamental to determining the circle's size and area.
The formula \( A(r) = \pi r^2 \) shows that the area is directly affected by the square of the radius:

• The larger the radius, the larger the square portion \( (r^2) \), hence a larger area.
• Since the radius defines dimensions in other circular-related expressions (like circumference), it’s crucial in various geometrical contexts.
• It simplifies calculations: By understanding the relationship between smaller radii and squares, estimations become intuitive.

Grasping what the radius means allows for a better understanding of how it influences the size and other properties of circular shapes.