# Writing equations for rate of change problems

• Poppynz
In summary, the conversation is about writing an equation for finding the rate of change of angle of a telescope that is 75m above water level on a cliff, as a boat approaches at 6m/s. The suggested equation is y=arctan(75/x), but it is pointed out that the correct form is tanθ = 75/x. The conversation ends with a suggestion to differentiate both sides using the chain rule.
Poppynz
Hi
Im trying to write an equation for the question below. i did write one but am pretty sure it is wrong because I need to differentiate arctan in it and we have not been taught that yet. Could someone please point me in the right direction with writing it?

## Homework Statement

A telescope is 75m above water level on a cliff and a boat is approaching at 6m/s. what is the rate of change of angle of the telescope when the boat is 75m from shore.

## The Attempt at a Solution

I thought it might be y=arctan(75/x) because the angle specified is found using arctan(opposite/addjacent).

Any suggestions appreciated :)

Poppynz said:
… I thought it might be y=arctan(75/x) because the angle specified is found using arctan(opposite/addjacent).

Any suggestions appreciated :)

Hi Poppynz!

Just write it tanθ = 75/x, and differentiate both sides wrt x

from the chain rule, you'll get a sec2θ, which you can rewrite in terms of the original 75/x

Thanks that seems much easier :)

## 1. What is the formula for calculating rate of change?

The formula for calculating rate of change is change in y divided by change in x. Mathematically, it can be written as (y2 - y1) / (x2 - x1).

## 2. How do I write an equation for a rate of change problem?

To write an equation for a rate of change problem, you first need to identify the variables involved and assign them to the correct positions in the formula. Then, substitute the given values for each variable and solve the equation to find the rate of change.

## 3. What units should I use for the rate of change?

The units for the rate of change will depend on the units of the variables involved in the problem. Make sure to use the same units for both the numerator and denominator when calculating the rate of change.

## 4. How is rate of change different from slope?

Rate of change and slope are closely related concepts, but they are not the same. Rate of change represents the change in one variable with respect to another, while slope represents the change in y over the change in x for a straight line. However, the formula for both is the same: change in y divided by change in x.

## 5. Can rate of change be negative?

Yes, rate of change can be negative. A negative rate of change means that the dependent variable is decreasing as the independent variable increases. It is important to pay attention to the context of the problem to determine the meaning of a negative rate of change.

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