A school wishes to form three sides of a rectangular playground using 320 meters of fencing. The playground borders the school building, so the fourth side does not need fencing. As shown below, one of the sides has length x (in meters). Side along school building (a) Find a function that gives the area A(x) of the playground (in square meters) in terms of x. A(:) = 0
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.
In the diagram:
- The horizontal side along the school building is labeled \( x \) meters.
- The playground's layout shown is rectangular with the school side.
#### Questions:
(a) **Find a function that gives the area \( A(x) \) of the playground (in square meters) in terms of \( x \).**
\[ A(x) = \]
(b) **What side length \( x \) gives the maximum area that the playground can have?**
\[
\text{Side length } x: \quad \boxed{\text{ meters}}
\]
(c) **What is the maximum area that the playground can have?**
\[
\text{Maximum area:} \quad \boxed{\text{ square meters}}
\]
### Diagram Explanation:
The diagram illustrates the rough layout of the playground with:
- The side along the school building labeled \( x \).
- Two sides perpendicular to the school building that are each shorter than 320 meters of the available fencing because fencing is required on three sides.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8dbcbf0b-9123-4a1c-9796-89fa34227664%2F9169ef92-d8c2-4ce0-b6db-eb7e8b453ffd%2Fuib5w8in_processed.jpeg&w=3840&q=75)
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