Mathematical Formulas and Theorems Used

Oct 13, 2025 by talynt.org 12 views

Introduction

In this problem, we are given two buildings on opposite sides of a highway. The distance between the two buildings is represented by the polynomial $3x^3 - x^2 + 7x + 100$ . One building is $2x^2 + 7x$ feet from the highway, and the other building is $x^3 + 2x^2 - 18$ feet from the highway. Our goal is to find the distance between the two buildings.

Step 1: Understand the Problem

To solve this problem, we need to understand the concept of distance and how it relates to the given polynomials. The distance between two points is the absolute value of the difference between their coordinates. In this case, the coordinates are represented by the polynomials $3x^3 - x^2 + 7x + 100$ and $x^3 + 2x^2 - 18$ .

Step 2: Find the Distance Between the Two Buildings

To find the distance between the two buildings, we need to subtract the polynomial representing the distance from the highway to one building from the polynomial representing the distance from the highway to the other building. This will give us the distance between the two buildings.

Let's denote the polynomial representing the distance from the highway to one building as $d_1(x)$ and the polynomial representing the distance from the highway to the other building as $d_2(x)$ . Then, the distance between the two buildings is given by:

d(x) = d_2(x) - d_1(x)

Substituting the given polynomials, we get:

d(x) = (x^3 + 2x^2 - 18) - (2x^2 + 7x)

Simplifying the expression, we get:

d(x) = x^3 + 2x^2 - 18 - 2x^2 - 7x

Combine like terms:

d(x) = x^3 - 7x - 18

Step 3: Simplify the Expression

To simplify the expression, we can factor out the greatest common factor (GCF) of the terms. In this case, the GCF is 1, so we cannot factor out any common factors.

However, we can rewrite the expression in a more convenient form by combining the like terms:

d(x) = x^3 - 7x - 18

Step 4: Find the Correct Answer

The correct answer is the polynomial $x^3 - 7x - 18$ .

Conclusion

In this problem, we were given two buildings on opposite sides of a highway. The distance between the two buildings is represented by the polynomial $3x^3 - x^2 + 7x + 100$ . One building is $2x^2 + 7x$ feet from the highway, and the other building is $x^3 + 2x^2 - 18$ feet from the highway. We found the distance between the two buildings by subtracting the polynomial representing the distance from the highway to one building from the polynomial representing the distance from the highway to the other building. The correct answer is the polynomial $x^3 - 7x - 18$ .

Mathematical Formulas and Theorems Used

Distance Formula: The distance between two points is the absolute value of the difference between their coordinates.
Polynomial Subtraction: To find the distance between two buildings, we need to subtract the polynomial representing the distance from the highway to one building from the polynomial representing the distance from the highway to the other building.

Real-World Applications

This problem has real-world applications in various fields, such as:

Civil Engineering: In civil engineering, we need to calculate the distance between buildings to design and construct safe and efficient structures.
Surveying: In surveying, we need to calculate the distance between landmarks to create accurate maps and plans.
Geography: In geography, we need to calculate the distance between locations to understand the relationships between different places.

Limitations and Future Work

This problem has some limitations, such as:

Assumptions: We assumed that the buildings are on opposite sides of the highway, which may not always be the case.
Simplifications: We simplified the expression by combining like terms, which may not always be possible.

Future work could include:

More Complex Problems: We could create more complex problems that involve multiple buildings and highways.
Real-World Data: We could use real-world data to create more accurate and realistic problems.

Conclusion

Q: What is the distance between the two buildings?

A: The distance between the two buildings is represented by the polynomial $x^3 - 7x - 18$ .

Q: How did you find the distance between the two buildings?

A: We found the distance between the two buildings by subtracting the polynomial representing the distance from the highway to one building from the polynomial representing the distance from the highway to the other building.

Q: What are the polynomials representing the distance from the highway to one building and the other building?

A: The polynomial representing the distance from the highway to one building is $2x^2 + 7x$ , and the polynomial representing the distance from the highway to the other building is $x^3 + 2x^2 - 18$ .

Q: How do you subtract polynomials?

A: To subtract polynomials, we need to subtract the corresponding terms of the two polynomials. In this case, we subtracted the polynomial $2x^2 + 7x$ from the polynomial $x^3 + 2x^2 - 18$ .

Q: What is the greatest common factor (GCF) of the terms in the polynomial $x^3 - 7x - 18$ ?

A: The greatest common factor (GCF) of the terms in the polynomial $x^3 - 7x - 18$ is 1.

Q: Can you factor out any common factors from the polynomial $x^3 - 7x - 18$ ?

A: No, we cannot factor out any common factors from the polynomial $x^3 - 7x - 18$ .

Q: What are some real-world applications of this problem?

A: Some real-world applications of this problem include:

Civil Engineering: In civil engineering, we need to calculate the distance between buildings to design and construct safe and efficient structures.
Surveying: In surveying, we need to calculate the distance between landmarks to create accurate maps and plans.
Geography: In geography, we need to calculate the distance between locations to understand the relationships between different places.

Q: What are some limitations of this problem?

A: Some limitations of this problem include:

Assumptions: We assumed that the buildings are on opposite sides of the highway, which may not always be the case.
Simplifications: We simplified the expression by combining like terms, which may not always be possible.

Q: What are some future work ideas for this problem?

A: Some future work ideas for this problem include:

More Complex Problems: We could create more complex problems that involve multiple buildings and highways.
Real-World Data: We could use real-world data to create more accurate and realistic problems.

Q: How can I apply this problem to my own life?

A: You can apply this problem to your own life by thinking about real-world scenarios where you need to calculate distances between objects. For example, you might need to calculate the distance between two buildings to design a safe and efficient structure, or you might need to calculate the distance between landmarks to create an accurate map.

Conclusion

In conclusion, this problem involves finding the distance between two buildings on opposite sides of a highway. We used polynomial subtraction to find the correct answer, which is the polynomial $x^3 - 7x - 18$ . This problem has real-world applications in various fields and has some limitations and future work.