# Related rates and implicit differentiation

In summary, the plane is flying at a constant speed of 300 km/h, passes over a radar station at an altitude of 1 km, and then begins to climb at a 30 degree angle. The rate at which the distance between the plane and the radar station is increasing a minute later can be found using the Pythagorean theorem and differentiating to solve for the rate.

## Homework Statement

A plane flying with a constant speed of 300 km/h passes over a ground radar station at an altitude of 1 km and climbs at an angle of 30 degrees. At what rate is the distance from the plane to the radar station increasing a minute later?

a2+b2=c2

## The Attempt at a Solution

I got to (y+1)2+x2=d22

It's that extra kilometer that's bothering me. It gives the answer as approx 296 km/h

d1 is the distance between the starting and current point (5 km) and d1' is 300 km/h

I don't know how to get from d1 to d2

Of course, right as I post it I figure it out.

x2+y2=d12

Sub in, differentiate, solve. I'm going to leave it up, just for anyone who wants to check it out. Silly me...

## What is the difference between related rates and implicit differentiation?

Related rates involve finding the rate of change of one variable with respect to another variable, while implicit differentiation involves finding the derivative of an implicit function where both variables are present in the equation.

## How can I identify when to use related rates or implicit differentiation?

Related rates are typically used when the variables in the problem are changing at different rates and are related to each other through an equation. Implicit differentiation is used when the variables are both present in the equation and cannot be easily isolated.

## What is the process for solving related rates problems?

The process for solving related rates problems involves identifying the variables involved, writing an equation that relates the variables, differentiating the equation with respect to time, plugging in known values, and solving for the unknown rate.

## Can implicit differentiation be used for any type of function?

Yes, implicit differentiation can be used for any type of function, including trigonometric, exponential, and logarithmic functions. However, it may be more difficult to solve for the derivative in some cases.

## How can I check if my answer to a related rates or implicit differentiation problem is correct?

You can check your answer by plugging in the known values and your calculated rate of change into the original equation and making sure it satisfies the equation. You can also compare your answer to the answer in the back of the textbook or use an online calculator to verify your solution.

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