Problem 70 - Intermediate Algebra

Question

The equation \(A(r)=\pi r^{2}\) expresses the area of a circular region as a function of the length of a radius \((r)\). Use \(3.14\) as an approximation for \(\pi\), and compute \(A(2)\), \(A(3), A(12)\), and \(A(17)\).

Step-by-Step Solution

Answer

The areas are 12.56, 28.26, 452.16, and 907.46 for radii 2, 3, 12, and 17 respectively.

Step 1: Understand the Function

The function given is \(A(r) = \pi r^{2}\). This means the area \(A\) of a circle is calculated by multiplying \(\pi\) (approximately 3.14) by the square of the radius \(r\).

Step 2: Calculate \(A(2)\)

Substitute \(r = 2\) into the area function: \(A(2) = 3.14 \times 2^2\). Compute the square of the radius: \(2^2 = 4\). Then multiply by \(3.14\): \(A(2) = 3.14 \times 4 = 12.56\).

Step 3: Calculate \(A(3)\)

Substitute \(r = 3\) into the function: \(A(3) = 3.14 \times 3^2\). Compute the square of the radius: \(3^2 = 9\). Then multiply by \(3.14\): \(A(3) = 3.14 \times 9 = 28.26\).

Step 4: Calculate \(A(12)\)

For \(r = 12\), substitute into the function: \(A(12) = 3.14 \times 12^2\). Calculate the square: \(12^2 = 144\). Then, \(A(12) = 3.14 \times 144 = 452.16\).

Step 5: Calculate \(A(17)\)

With \(r = 17\), substitute into the function: \(A(17) = 3.14 \times 17^2\). Compute the square: \(17^2 = 289\). Then, \(A(17) = 3.14 \times 289 = 907.46\).

Key Concepts

Function Evaluation

It all starts with understanding what a function is in mathematics. A function is like a machine that takes an input, processes it, and provides an output. In the context of calculating the area of a circle, the formula is given as a function: \(A(r) = \pi r^2\). Here, \(A(r)\) is the function where:

• \(A\) stands for area.
• \(r\) represents the radius, which is the input to our function.
• \(\pi\), a mathematical constant, is involved in the process of calculating the output.

To evaluate a function means to find the output for a specific input. For example, if the input is \(r = 2\), we will substitute 2 into the function to find \(A(2)\). Each distinct value of \(r\) provides a different area based on this function.

Substitution

One of the core techniques in solving functions like \(A(r) = \pi r^2\) is by using substitution. Substitution is a straightforward process where the given input value (in this case, the radius \(r\)) is placed into the function in place of \(r\).Let's break this down:

• Start with the general formula \(A(r) = \pi r^2\).
• Pick the radius you need to compute the area for, such as \(r = 2\).
• Substitute 2 for \(r\) in the function: \(A(2) = \pi \times 2^2\).

After substituting, you compute the result of \(2^2\), and then multiply it by \(\pi\). This same process is followed for other values like \(r = 3\), \(r = 12\), and \(r = 17\), showcasing how substitution allows you to evaluate functions for any given values.

Mathematical Constants

In mathematics, a constant is a number that doesn’t change. Constants are vital in computations, and one of the most iconic is \(\pi\), which is approximately 3.14. In the formula for the area of a circle, \(\pi\) is crucial because it helps relate the radius to the area. Without constants, mathematical formulas would be incomplete. Here’s why \(\pi\) is important:

• \(\pi\) is used to calculate circular properties such as area and circumference.
• It's a part of the formula because of how circles inherently relate to the number 3.14.
• Approximating \(\pi\) as 3.14 helps simplify calculations without sacrificing too much accuracy.

When using constants, you usually substitute approximate values to make computations easier and results comprehensible. This approximation is especially helpful in practical situations where precise decimal values aren't necessary. Hence, understanding the role of constants like \(\pi\) enables smoother mathematical processes.