Problem 46
Question
Use numerical evaluation on the equations. Geometry (area of a circle) \(A=\pi r^{2}\). Find \(A\) if \(\pi\) is approximately 3.14 and \(r=11\).
Step-by-Step Solution
Verified Answer
Answer: The area of the circle is approximately 380.34 square units.
1Step 1: Identify the formula for the area of a circle
The formula to calculate the area of a circle is given by: \(A=\pi r^{2}\), where \(A\) is the area, \(\pi\) is a mathematical constant approximately equal to 3.14, and \(r\) is the radius of the circle.
2Step 2: Determine the given values
For this exercise, we are given the radius of the circle, \(r=11\), and the approximate value of \(\pi\) as 3.14.
3Step 3: Calculate the area of the circle
Now, all we need to do is substitute the given values into the formula and perform the calculation:
\(A = 3.14 \times (11)^{2}\)
\(A = 3.14 \times 121\)
\(A ≈ 380.34\)
So, the area of the circle is approximately 380.34 square units.
Key Concepts
Numerical EvaluationGeometryCircle Formula
Numerical Evaluation
Numerical evaluation is a fundamental aspect of mathematics, particularly useful when dealing with formulas and equations that involve constants and variables. At its core, numerical evaluation involves substituting specific numbers in place of symbols in an equation and then performing the indicated operations. For example, if you're given the equation for the area of a circle, you'll need to replace the variable representing the radius and the constant \( \pi \) with numerical values to calculate the area.
To ensure accuracy during numerical evaluation, follow these steps:
To ensure accuracy during numerical evaluation, follow these steps:
- Identify the constants and variables in the equation.
- Substitute the known numerical values for the constants and variables.
- Perform the arithmetic operations following the correct order of operations, which is to first handle exponents, then multiplication and division, followed by addition and subtraction.
Geometry
Geometry is a branch of mathematics focused on the properties and relations of points, lines, surfaces, and solids. In geometry, understanding the properties of shapes is crucial, as it allows you to apply formulas like those for calculating the area of a circle correctly. The area is a measure of the amount of space a two-dimensional shape covers. It's typically measured in square units, such as square meters or square centimeters.
Often seen as the study of shapes and space, geometry can be visualized in the world around us, from the design of a sports field to the structure of a building. When you look at a circular object, like a wheel or a pizza, you're observing one of the fundamental shapes in geometry. In the textbook solution, we're using the understanding of geometric principles to evaluate the area covered by a shape with a constant distance from a central point—a circle.
Often seen as the study of shapes and space, geometry can be visualized in the world around us, from the design of a sports field to the structure of a building. When you look at a circular object, like a wheel or a pizza, you're observing one of the fundamental shapes in geometry. In the textbook solution, we're using the understanding of geometric principles to evaluate the area covered by a shape with a constant distance from a central point—a circle.
Circle Formula
The circle formula \( A = \pi r^{2} \) is a simple yet powerful expression used to calculate the area of a circle. It states that the area \( A \) of a circle is equal to the mathematical constant \( \pi \) multiplied by the square of the circle's radius \( r \). The radius is the distance from the center of the circle to any point on its edge. Because the circle is a symmetrical shape, this distance is consistent no matter which direction from the center you measure.
Understanding this formula is crucial for solving geometric problems involving circles. In practice, you'll usually use the approximate value for pi \( \pi \approx 3.14 \), although more precise calculators will have a pi button for better accuracy. Substituting into the circle formula, as we did in the step-by-step solution, and squaring the radius give you the area, provided you perform the calculations correctly. This process illustrates how geometry and numerical evaluation come together to solve practical mathematical problems, like finding the amount of material needed to craft a circular table top or determining the size of a round garden.
Understanding this formula is crucial for solving geometric problems involving circles. In practice, you'll usually use the approximate value for pi \( \pi \approx 3.14 \), although more precise calculators will have a pi button for better accuracy. Substituting into the circle formula, as we did in the step-by-step solution, and squaring the radius give you the area, provided you perform the calculations correctly. This process illustrates how geometry and numerical evaluation come together to solve practical mathematical problems, like finding the amount of material needed to craft a circular table top or determining the size of a round garden.
Other exercises in this chapter
Problem 46
For the following problems, simplify each of the algebraic expressions. $$ 4 x^{0}+3 x^{0}-5 x^{0}+7 x^{0}-x^{0}, \quad x \neq 0 $$
View solution Problem 46
Classify each of the equations for the following problems by degree. If the term linear, quadratic, or cubic applies, state it. $$ 8 a+2 b=4 b-8 $$
View solution Problem 46
For the following problems, perform the multiplications and combine any like terms. $$ -7(4 x+2) $$
View solution Problem 46
For the following problems, list, if any should appear, the common factors in the expressions. $$ 15 x^{2}-30 x y^{2} $$
View solution