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.
![### Geometry Problem: Finding the Area of the Quadrilateral
#### Problem Statement
**Question 15:** Find the area. *Note, PQ = 13 and 12 is 1/2 of the diagonal PR.*
#### Diagram Description
- A quadrilateral \( PQRS \) is shown in the diagram.
- Diagonal \( PR \) is indicated.
- \( PQ \) has a length of 13 meters.
- The length of 12 meters is marked as half of the diagonal \( PR \).
- The quadrilateral is divided into two triangles by the diagonal \( PR \).
#### Steps to Find the Area
1. **Understanding the Diagram:**
- Given that 12 meters is half of diagonal \( PR \), the full length of \( PR \) is \( 2 \times 12 = 24 \) meters.
- Therefore, \( PR = 24 \) meters.
2. **Area Calculation of Triangles:**
- Using the fact that \( PQRS \) is divided into two triangles by diagonal \( PR \), calculate the areas of these triangles separately and sum them up.
3. **Breakdown:**
- Calculate the area of triangle \( PQS \):
- Use the given side lengths and any geometric properties.
- Calculate the area of triangle \( PRS \):
- Use the given side lengths and any geometric properties.
4. **Formula Application:**
- For a simple calculation, we may need the heights corresponding to these bases \( PQ \) and \( PR \).
- Use appropriate geometric or trigonometric methods to find heights, and apply the triangle area formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
#### Answer Choices
- Based on the problem's solution, choose or match the correct area (in square meters) from the following options:
\[
\begin{array}{c}
\text{(a)} \, 120 \\
\text{(b)} \, 10 \\
\text{(c)} \, 24 \\
\text{(d)} \, 52 \\
\text{(e)} \, 240 \\
\end{array}
\]
#### Note
Ensure all steps align with the mathematical principles and validate](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc922fe94-7b6a-4ac6-bdc3-3ec01d0b9c76%2Fc4af7dc4-4fc3-40dc-9561-e83ca1c3f7af%2Fgcjhefk_processed.jpeg&w=3840&q=75)
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