Problem 10 - Algebra A Combined Function

Question

Substitute the given values into each given formula and solve for the unknown variable. $$ A=\pi r^{2} ; \quad r=4 $$

Step-by-Step Solution

Answer

The area is \( 16\pi \) or approximately 50.24 using \( \pi \approx 3.14 \).

Step 1: Substitute the Given Value

First, identify the variables in the formula. Here, the formula is for the area of a circle: \( A = \pi r^2 \), where \( r = 4 \) is given. Substitute \( r = 4 \) into the formula, replacing \( r \) with \( 4 \). This gives us \( A = \pi (4)^2 \).

Step 2: Compute the Squared Term

Calculate \( 4^2 \), the squared term in the formula. \( 4^2 = 16 \). Now the formula looks like this: \( A = \pi imes 16 \).

Step 3: Multiply by Pi

Next, multiply \( 16 \) by \( \pi \) to find the area. The area \( A \) becomes \( A = 16\pi \), which is the final answer using \( \pi \) as a symbol. Note: If you need a decimal approximation, you can use \( \pi \approx 3.14 \) to compute \( 16 \times 3.14 = 50.24 \).

Key Concepts

Area of a Circle

The area of a circle is a concept that tells us the amount of space inside the circle's boundary. Imagine trying to fill a circle with little tiles; the area would tell you how many tiles you'd need.
For a circle, the formula to find this is:

• \( A = \pi r^2 \)

In this formula, \( A \) stands for the area, \( \pi \) (pi) is a special mathematical constant, and \( r \) is the radius of the circle.
The radius is the distance from the center of the circle to any point on its edge. Substituting the radius value into the formula and solving for \( A \) allows us to calculate how much space is inside the circle.

You substitute the value of \( r \) and then perform the exponentiation operation to solve \( r^2 \), finally multiplying by \( \pi \), which gives you the area.

Pi (π)

Have you ever wondered about the mysterious \( \pi \) in mathematical formulas? \( \pi \) (pronounced "pie") is a crucial constant in mathematics, especially when dealing with circles. It represents the ratio of a circle's circumference to its diameter. No matter the size of the circle, this ratio always equals approximately 3.14159.

Here are some key facts about \( \pi \):

• It's an irrational number, meaning it can't be exactly expressed as a simple fraction.
• It's often approximated as 3.14 for ease of calculation.
• \( \pi \) is used in formulas not just for the area of a circle, but also for the circumference, the surface area of a sphere, and more.

In the area of a circle problem, \( \pi \) is the factor we multiply after calculating \( r^2 \), showing how it scales up the area calculation to fit the actual circle.

Exponentiation

Exponentiation might sound complex, but it simply refers to the operation of multiplying a number by itself a certain number of times. When you see \( r^2 \), it means "multiply \( r \) by itself once."

Here's what happens step-by-step:

• If \( r = 4 \), then \( r^2 \) means \( 4 \times 4 \).
• This results in 16, completing the "squared" part of the formula.

This mathematical operation is part of the key steps in finding a circle's area. Computing \( r^2 \) helps us understand how much of the area is multiplied by \( \pi \), completing our work to understand the circle's full area.

The power of two tells us we're dealing with a shape in two dimensions, like the flat space inside a circle.