What is the simplest form of 7/18? What is the rational form of 1/3? Which of the following is the simplest form of 16a^3b^3? What is the sum of 2√3 and 4√3? Which of the following... What is the simplest form of 7/18? What is the rational form of 1/3? Which of the following is the simplest form of 16a^3b^3? What is the sum of 2√3 and 4√3? Which of the following statements is not true? What is the first step to do to solve √x + 2 = 6? Which of the following is the value of x on number 34? What is the distance between the airplane and the helicopter?
Understand the Problem
The question set consists of various math problems that require simplification of radicals, evaluation of expressions, and understanding geometric relationships between a plane and a helicopter. The user is likely looking for solutions to these problems or clarification of the concepts involved.
Answer
The distance between the airplane and the helicopter is $10$ miles.
Answer for screen readers
The distance between the airplane and the helicopter is $10$ miles.
Steps to Solve
-
Identify the problem type The problem involves simplifying and evaluating expressions, including radicals. Additionally, it relates to geometric relationships regarding distances and directions.
-
Simplifying radicals Start with a simpler expression, such as simplifying $\sqrt{75}$. Since $75 = 25 \cdot 3$, we have: $$ \sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3} $$
-
Rationalizing expressions When dealing with expressions like $\frac{1}{\sqrt{3}}$, multiply the numerator and denominator by $\sqrt{3}$ to obtain: $$ \frac{\sqrt{3}}{3} $$
-
Evaluating expressions For example, to evaluate $2 + \sqrt{2}$, calculate the value of $\sqrt{2} \approx 1.414$, leading to: $$ 2 + \sqrt{2} \approx 3.414 $$
-
Finding the distance between two points Using the distance formula for points $(x_1, y_1)$ and $(x_2, y_2)$: $$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$ If the plane is at $(8, 0)$ and the helicopter is at $(0, 6)$, then: $$ d = \sqrt{(8 - 0)^2 + (0 - 6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ miles} $$
-
Concluding steps Check that the distance corresponds to the calculated coordinates based on their displacement.
The distance between the airplane and the helicopter is $10$ miles.
More Information
This problem illustrates the application of the Pythagorean theorem in determining distances in a coordinate system, where the plane and helicopter travel in perpendicular directions from a common point.
Tips
- Misapplying the Pythagorean theorem by not recognizing the coordinates correctly.
- Forgetting to simplify square roots properly.
- Failing to rationalize expressions when required.
AI-generated content may contain errors. Please verify critical information
