The Function $f(x)=-(x-3)^2+9$ Can Be Used To Represent The Area Of A Rectangle With A Perimeter Of 12 Units, As A Function Of The Length Of The Rectangle, $x$. What Is The Maximum Area Of The Rectangle?A. 3 Square Units B. 6

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Understanding the Problem

The given function, $f(x)=-(x-3)^2+9$, represents the area of a rectangle with a perimeter of 12 units, as a function of the length of the rectangle, $x$. To find the maximum area of the rectangle, we need to analyze the given function and understand its behavior.

Analyzing the Function

The given function is a quadratic function in the form of $f(x)=a(x-h)^2+k$, where $a=-1$, $h=3$, and $k=9$. The graph of this function is a parabola that opens downwards, with its vertex at the point $(h, k) = (3, 9)$.

Interpreting the Function

Since the function represents the area of a rectangle with a perimeter of 12 units, we can interpret the function as follows:

  • The length of the rectangle is represented by the variable $x$.
  • The area of the rectangle is represented by the function $f(x)=-(x-3)^2+9$.
  • The maximum area of the rectangle occurs at the vertex of the parabola, which is at the point $(h, k) = (3, 9)$.

Finding the Maximum Area

To find the maximum area of the rectangle, we need to find the value of the function at the vertex of the parabola. Since the vertex is at the point $(h, k) = (3, 9)$, we can substitute $x=3$ into the function to find the maximum area:

f(3)=−(3−3)2+9=9f(3)=-(3-3)^2+9=9

Therefore, the maximum area of the rectangle is 9 square units.

Conclusion

In conclusion, the function $f(x)=-(x-3)^2+9$ represents the area of a rectangle with a perimeter of 12 units, as a function of the length of the rectangle, $x$. By analyzing the function and interpreting its behavior, we found that the maximum area of the rectangle occurs at the vertex of the parabola, which is at the point $(h, k) = (3, 9)$. Therefore, the maximum area of the rectangle is 9 square units.

The Importance of Understanding Quadratic Functions

Understanding quadratic functions is crucial in mathematics, as they are used to model a wide range of real-world phenomena. In this problem, the quadratic function represents the area of a rectangle with a perimeter of 12 units, as a function of the length of the rectangle. By analyzing the function and interpreting its behavior, we can find the maximum area of the rectangle.

Real-World Applications of Quadratic Functions

Quadratic functions have numerous real-world applications, including:

  • Physics: Quadratic functions are used to model the motion of objects under the influence of gravity, friction, and other forces.
  • Engineering: Quadratic functions are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
  • Economics: Quadratic functions are used to model the behavior of economic systems, including supply and demand curves.

Conclusion

In conclusion, the function $f(x)=-(x-3)^2+9$ represents the area of a rectangle with a perimeter of 12 units, as a function of the length of the rectangle, $x$. By analyzing the function and interpreting its behavior, we found that the maximum area of the rectangle occurs at the vertex of the parabola, which is at the point $(h, k) = (3, 9)$. Therefore, the maximum area of the rectangle is 9 square units.

Final Answer

The final answer is: 9\boxed{9}

Understanding the Problem

The given function, $f(x)=-(x-3)^2+9$, represents the area of a rectangle with a perimeter of 12 units, as a function of the length of the rectangle, $x$. To find the maximum area of the rectangle, we need to analyze the given function and understand its behavior.

Q&A

Q: What is the given function?

A: The given function is $f(x)=-(x-3)^2+9$, which represents the area of a rectangle with a perimeter of 12 units, as a function of the length of the rectangle, $x$.

Q: What is the maximum area of the rectangle?

A: The maximum area of the rectangle is 9 square units.

Q: How do we find the maximum area of the rectangle?

A: To find the maximum area of the rectangle, we need to analyze the function and interpret its behavior. We can find the maximum area by substituting $x=3$ into the function.

Q: What is the significance of the vertex of the parabola?

A: The vertex of the parabola represents the maximum area of the rectangle. In this case, the vertex is at the point $(h, k) = (3, 9)$.

Q: What are some real-world applications of quadratic functions?

A: Quadratic functions have numerous real-world applications, including:

  • Physics: Quadratic functions are used to model the motion of objects under the influence of gravity, friction, and other forces.
  • Engineering: Quadratic functions are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
  • Economics: Quadratic functions are used to model the behavior of economic systems, including supply and demand curves.

Q: How do we interpret the function in the context of the problem?

A: We can interpret the function as follows:

  • The length of the rectangle is represented by the variable $x$.
  • The area of the rectangle is represented by the function $f(x)=-(x-3)^2+9$.
  • The maximum area of the rectangle occurs at the vertex of the parabola, which is at the point $(h, k) = (3, 9)$.

Q: What is the importance of understanding quadratic functions?

A: Understanding quadratic functions is crucial in mathematics, as they are used to model a wide range of real-world phenomena. In this problem, the quadratic function represents the area of a rectangle with a perimeter of 12 units, as a function of the length of the rectangle.

Conclusion

In conclusion, the function $f(x)=-(x-3)^2+9$ represents the area of a rectangle with a perimeter of 12 units, as a function of the length of the rectangle, $x$. By analyzing the function and interpreting its behavior, we found that the maximum area of the rectangle occurs at the vertex of the parabola, which is at the point $(h, k) = (3, 9)$. Therefore, the maximum area of the rectangle is 9 square units.

Final Answer

The final answer is: 9\boxed{9}