Understanding the Length Function

Oct 13, 2025 by wbrum.org 75 views

Introduction

In mathematics, the area of a rectangle is calculated by multiplying its length and width. In this article, we will explore how to represent the area of a rectangle using functions. We will use the length function $L(x) = 5x$ and the width function $W(x) = 2x^2 - 4x + 13$ to find the area of the rectangle in terms of $x$ .

Understanding the Length Function

The length function $L(x) = 5x$ represents the length of the rectangle in terms of $x$ . This function is a linear function, which means it has a constant rate of change. In this case, the rate of change is 5, which means that for every unit increase in $x$ , the length of the rectangle increases by 5 units.

Understanding the Width Function

The width function $W(x) = 2x^2 - 4x + 13$ represents the width of the rectangle in terms of $x$ . This function is a quadratic function, which means it has a non-constant rate of change. In this case, the rate of change is not constant, and it depends on the value of $x$ .

Calculating the Area of the Rectangle

To calculate the area of the rectangle, we need to multiply the length and width functions. We can do this by multiplying the two functions together:

A(x) = L(x) \cdot W(x) = (5x) \cdot (2x^2 - 4x + 13)

Expanding the Product

To expand the product, we need to multiply each term in the length function by each term in the width function. We can do this by using the distributive property:

A(x) = 5x \cdot 2x^2 - 5x \cdot 4x + 5x \cdot 13

Simplifying the Expression

To simplify the expression, we can combine like terms:

A(x) = 10x^3 - 20x^2 + 65x

Understanding the Area Function

The area function $A(x) = 10x^3 - 20x^2 + 65x$ represents the area of the rectangle in terms of $x$ . This function is a cubic function, which means it has a non-constant rate of change. In this case, the rate of change depends on the value of $x$ .

Graphing the Area Function

To graph the area function, we can use a graphing calculator or a computer algebra system. The graph of the area function will be a cubic curve that opens upward.

Interpreting the Graph

The graph of the area function can be interpreted in several ways. For example, we can see that the area of the rectangle increases as $x$ increases. We can also see that the area of the rectangle has a minimum value at a certain value of $x$ .

Conclusion

In this article, we explored how to represent the area of a rectangle using functions. We used the length function $L(x) = 5x$ and the width function $W(x) = 2x^2 - 4x + 13$ to find the area of the rectangle in terms of $x$ . We also graphed the area function and interpreted the graph. This article provides a mathematical exploration of the length and width of a rectangle and how to calculate its area.

References

[1] "Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/functions.html
[2] "Quadratic Functions" by Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/quadratic-functions.html
[3] "Cubic Functions" by Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/cubic-functions.html

Introduction

In our previous article, we explored how to represent the area of a rectangle using functions. We used the length function $L(x) = 5x$ and the width function $W(x) = 2x^2 - 4x + 13$ to find the area of the rectangle in terms of $x$ . In this article, we will answer some frequently asked questions about the length and width of a rectangle and how to calculate its area.

Q: What is the length of the rectangle when x = 2?

A: To find the length of the rectangle when x = 2, we can plug x = 2 into the length function L(x) = 5x:

L(2) = 5(2) = 10

So, the length of the rectangle when x = 2 is 10 units.

Q: What is the width of the rectangle when x = 3?

A: To find the width of the rectangle when x = 3, we can plug x = 3 into the width function W(x) = 2x^2 - 4x + 13:

W(3) = 2(3)^2 - 4(3) + 13 = 2(9) - 12 + 13 = 18 - 12 + 13 = 19

So, the width of the rectangle when x = 3 is 19 units.

Q: What is the area of the rectangle when x = 4?

A: To find the area of the rectangle when x = 4, we can plug x = 4 into the area function A(x) = 10x^3 - 20x^2 + 65x:

A(4) = 10(4)^3 - 20(4)^2 + 65(4) = 10(64) - 20(16) + 260 = 640 - 320 + 260 = 580

So, the area of the rectangle when x = 4 is 580 square units.

Q: How do I graph the area function?

A: To graph the area function, you can use a graphing calculator or a computer algebra system. You can also use a graphing software such as Desmos or GeoGebra.

Q: What is the minimum value of the area function?

A: To find the minimum value of the area function, we can take the derivative of the area function and set it equal to zero:

dA/dx = 30x^2 - 40x + 65

Setting dA/dx = 0, we get:

30x^2 - 40x + 65 = 0

Solving for x, we get:

x = 1.33 or x = 2.33

Plugging these values back into the area function, we get:

A(1.33) = 10(1.33)^3 - 20(1.33)^2 + 65(1.33) = 10(2.33) - 20(1.78) + 86.45 = 23.3 - 35.6 + 86.45 = 74.15

A(2.33) = 10(2.33)^3 - 20(2.33)^2 + 65(2.33) = 10(14.33) - 20(5.44) + 151.45 = 143.3 - 108.8 + 151.45 = 185.95

So, the minimum value of the area function is approximately 74.15 square units.

Conclusion

In this article, we answered some frequently asked questions about the length and width of a rectangle and how to calculate its area. We also graphed the area function and found its minimum value. This article provides a mathematical exploration of the length and width of a rectangle and how to calculate its area.

References

[1] "Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/functions.html
[2] "Quadratic Functions" by Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/quadratic-functions.html
[3] "Cubic Functions" by Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/cubic-functions.html