An Incoming Airplane Is $x$ Miles Due North From The Control Tower At An Airport. A Second Incoming Airplane Is $y$ Miles Due East Of The Same Control Tower. The Shortest Distance Between The Two Airplanes Is $z$ Miles.Which

Introduction

In this article, we will delve into a classic problem in mathematics that involves the shortest distance between two points in a two-dimensional plane. The problem is as follows: an incoming airplane is $x$ miles due north from the control tower at an airport, and a second incoming airplane is $y$ miles due east of the same control tower. The shortest distance between the two airplanes is $z$ miles. Our goal is to find the value of $z$ in terms of $x$ and $y$.

The Problem

To approach this problem, we need to visualize the situation on a coordinate plane. Let's assume that the control tower is located at the origin $(0, 0)$ of the coordinate plane. The first airplane is located at the point $(0, x)$, and the second airplane is located at the point $(y, 0)$. The shortest distance between the two airplanes is the length of the line segment connecting the two points.

Using the Pythagorean Theorem

One way to find the shortest distance between the two airplanes is to use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In this case, we can form a right-angled triangle with the two airplanes and the line segment connecting them as the hypotenuse. The length of the line segment is the shortest distance between the two airplanes, which we want to find. The lengths of the other two sides of the triangle are $x$ and $y$, which are the distances of the two airplanes from the control tower.

Using the Pythagorean theorem, we can write:

$$z^2 = x^2 + y^2$$

Solving for $z$

To find the value of $z$, we can take the square root of both sides of the equation:

$$z = \sqrt{x^2 + y^2}$$

This is the formula for the shortest distance between the two airplanes in terms of $x$ and $y$.

Geometric Interpretation

The formula $z = \sqrt{x^2 + y^2}$ can be interpreted geometrically as the distance between two points in a two-dimensional plane. The square root of the sum of the squares of the distances of the two points from the origin is equal to the distance between the two points.

Example

Let's consider an example to illustrate the formula. Suppose that the first airplane is 3 miles due north from the control tower, and the second airplane is 4 miles due east of the control tower. We can use the formula to find the shortest distance between the two airplanes:

$$z = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$

Therefore, the shortest distance between the two airplanes is 5 miles.

Conclusion

In this article, we have used the Pythagorean theorem to find the shortest distance between two points in a two-dimensional plane. The formula $z = \sqrt{x^2 + y^2}$ can be used to find the distance between two points in terms of their distances from the origin. We have also provided a geometric interpretation of the formula and an example to illustrate its use.

Applications

The formula $z = \sqrt{x^2 + y^2}$ has many applications in mathematics and science. It can be used to find the distance between two points in a two-dimensional plane, which is a fundamental concept in geometry and trigonometry. It can also be used to find the shortest distance between two points in a three-dimensional space, which is a fundamental concept in calculus and physics.

Further Reading

For further reading on this topic, we recommend the following resources:

• "Geometry: A Comprehensive Introduction" by Dan Pedoe
• "Trigonometry: A Unit Circle Approach" by Michael Corral
• "Calculus: Early Transcendentals" by James Stewart

References

• "The Pythagorean Theorem" by Michael S. Klamkin
• "Geometry: A Comprehensive Introduction" by Dan Pedoe
• "Trigonometry: A Unit Circle Approach" by Michael Corral

Appendix

The following is a list of formulas and theorems that are used in this article:

• Pythagorean theorem: $z^2 = x^2 + y^2$
• Formula for the shortest distance between two points: $z = \sqrt{x^2 + y^2}$
  Q&A: An Incoming Airplane Distance Problem =============================================

Introduction

In our previous article, we explored the problem of finding the shortest distance between two incoming airplanes. We used the Pythagorean theorem to derive the formula $z = \sqrt{x^2 + y^2}$, which gives the distance between the two airplanes in terms of their distances from the control tower. In this article, we will answer some common questions related to this problem.

Q: What is the significance of the Pythagorean theorem in this problem?

A: The Pythagorean theorem is a fundamental concept in geometry that relates the lengths of the sides of a right-angled triangle. In this problem, we use the theorem to find the shortest distance between the two airplanes, which is the length of the line segment connecting the two points.

Q: How do I apply the formula $z = \sqrt{x^2 + y^2}$ in real-life situations?

A: The formula $z = \sqrt{x^2 + y^2}$ can be used to find the distance between two points in a two-dimensional plane. For example, if you know the coordinates of two points, you can use the formula to find the distance between them. This is useful in a variety of situations, such as navigation, surveying, and engineering.

Q: What if the two airplanes are not on the same plane? How do I find the shortest distance between them?

A: If the two airplanes are not on the same plane, you need to use a different formula to find the shortest distance between them. In this case, you can use the formula for the distance between two points in three-dimensional space, which is given by:

$$z = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2}$$

where $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are the coordinates of the two points.

Q: Can I use the formula $z = \sqrt{x^2 + y^2}$ to find the distance between two points on a sphere?

A: No, the formula $z = \sqrt{x^2 + y^2}$ is only applicable to points in a two-dimensional plane. If you want to find the distance between two points on a sphere, you need to use a different formula that takes into account the curvature of the sphere.

Q: How do I handle cases where the two airplanes are at the same location?

A: If the two airplanes are at the same location, the distance between them is zero. In this case, the formula $z = \sqrt{x^2 + y^2}$ will give a result of zero, which is correct.

Q: Can I use the formula $z = \sqrt{x^2 + y^2}$ to find the distance between two points on a non-Euclidean surface?

A: No, the formula $z = \sqrt{x^2 + y^2}$ is only applicable to points on a Euclidean surface, such as a flat plane. If you want to find the distance between two points on a non-Euclidean surface, such as a sphere or a hyperbolic plane, you need to use a different formula that takes into account the curvature of the surface.

Conclusion

In this article, we have answered some common questions related to the problem of finding the shortest distance between two incoming airplanes. We have also provided some additional information on how to apply the formula $z = \sqrt{x^2 + y^2}$ in real-life situations.

Further Reading

For further reading on this topic, we recommend the following resources:

• "Geometry: A Comprehensive Introduction" by Dan Pedoe
• "Trigonometry: A Unit Circle Approach" by Michael Corral
• "Calculus: Early Transcendentals" by James Stewart

References

• "The Pythagorean Theorem" by Michael S. Klamkin
• "Geometry: A Comprehensive Introduction" by Dan Pedoe
• "Trigonometry: A Unit Circle Approach" by Michael Corral

Appendix

The following is a list of formulas and theorems that are used in this article:

• Pythagorean theorem: $z^2 = x^2 + y^2$
• Formula for the shortest distance between two points: $z = \sqrt{x^2 + y^2}$
• Formula for the distance between two points in three-dimensional space: $z = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2}$