Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
![### Question:
**Which is closest to the area of the shaded region of the circle?**
![Diagram of a circle with a shaded sector]
- The circle has a sector that is shaded.
- The central angle of the shaded sector is 72 degrees.
- The radius of the circle is 8 cm.
### Answer Choices:
- a) 10.1 cm²
- b) 50.3 cm²
- c) 160.8 cm²
- d) 98.4 cm²
- e) 40.2 cm²
### Diagram Explanation:
The diagram shows a circle with a radius of 8 cm. A sector of the circle is shaded, representing a central angle of 72 degrees.
### Calculation Steps:
To find the area of the shaded region (sector) of the circle, follow these steps:
1. **Area of the Entire Circle**: Use the formula \( A = \pi r^2 \).
\[
A = \pi \times (8 \text{ cm})^2 = 64\pi \text{ cm}^2
\]
2. **Proportion of the Circle Represented by the Shaded Sector**: The proportion is the fraction of the circle's 360 degrees that the sector represents.
\[
\frac{72^\circ}{360^\circ} = \frac{1}{5}
\]
3. **Area of the Shaded Sector**: Multiply the area of the entire circle by the proportion of the circle that is shaded.
\[
\text{Area of Shaded Sector} = \left(\frac{1}{5}\right) \times 64\pi \text{ cm}^2 \approx 40.2 \text{ cm}^2
\]
### Conclusion:
The area closest to the shaded region of the circle is:
- **e) 40.2 cm²**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2656de7-415a-4a39-958f-1d743e292993%2F691c32a0-4c00-4c4b-b623-be7282c5a699%2Fe63h8q_processed.jpeg&w=3840&q=75)
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