# Calculus airplane related rates problem ( cosine rule)

• MHB
• jaychay
In summary, the problem involves finding the rate of change of the distance between a student and an airplane, which is moving at a constant velocity of 120 meters per minute and at an angle of 60 degrees from the horizontal plane. The student is initially 5 meters away from the airplane and the problem asks for the rate of change when the airplane is 10 meters in the air. The solution involves using the distance formula, taking the derivative, and solving for the rate of change.
jaychay
A student has test his airplane and he is far from the airplane for 5 meter.He start to test his airplane by letting his airplane to move 60 degree from the horizontal plane with constant velocity for 120 meter per minute.Find the rate of distance between the student and the plane when the plane is moving 60 degree from the horizontal plane for 10 meter in the air ?

I have tried to solve the answer many times but I cannot do it
Thank you in advice

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Are you translating this to English from another language? I assume the problem wants to know the rate of change of the distance between the plane and student w/respect to time.

note 120 meters/min = 2 meters/sec

let $r$ be the distance between the student and the airplane at any time $t$ in seconds

$r^2 = (2t)^2 + 5^2 - 2(2t)(5)\cos(120^\circ)$

take the derivative of the above equation w/respect to time, then determine the value of $\dfrac{dr}{dt}$ in meters/sec

skeeter said:
Are you translating this to English from another language? I assume the problem wants to know the rate of change of the distance between the plane and student w/respect to time.

note 120 meters/min = 2 meters/sec

let $r$ be the distance between the student and the airplane at any time $t$ in seconds

$r^2 = (2t)^2 + 5^2 - 2(2t)(5)\cos(120^\circ)$

take the derivative of the above equation w/respect to time, then determine the value of $\dfrac{dr}{dt}$ in meters/sec
Thank you very much

## 1. How is the cosine rule used in airplane related rates problems?

The cosine rule is used to calculate the distance between two points on a triangle, which is often needed in airplane related rates problems. This distance can then be used to determine the rate of change of a certain variable, such as the angle of elevation or the distance traveled by the airplane.

## 2. Can you explain the steps for solving a calculus airplane related rates problem using the cosine rule?

To solve a calculus airplane related rates problem using the cosine rule, follow these steps:
1. Draw a diagram and label all known and unknown variables.
2. Use the cosine rule to find the distance between two points on the triangle.
3. Differentiate the equation with respect to time to find the rate of change of the desired variable.
4. Substitute in the known values and solve for the unknown rate of change.

## 3. What are some common mistakes to avoid when solving a calculus airplane related rates problem?

Some common mistakes to avoid when solving a calculus airplane related rates problem include:
- Forgetting to label the diagram correctly
- Using the wrong formula or equation
- Forgetting to differentiate the equation with respect to time
- Not substituting in the known values correctly
- Making calculation errors

## 4. How does the cosine rule relate to the Pythagorean theorem?

The Pythagorean theorem is a special case of the cosine rule, where one of the angles in the triangle is a right angle. The cosine rule can be used to find the length of the hypotenuse in a right triangle, which is equivalent to using the Pythagorean theorem.

## 5. Are there any real-life applications of using the cosine rule in airplane related rates problems?

Yes, the cosine rule is commonly used in real-life applications involving airplanes, such as calculating the rate of change of an airplane's altitude or the distance traveled by an airplane. It is also used in navigation and surveying to determine the distance between two points on a map or in the field.

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