# Exciting Dirt Removal Calculations!

## How can we determine the rate of change in the amount of dirt removed over time?

a) What does the derivative of the dirt amount with respect to time represent?

b) How does the integral of dirt amount over time help us calculate the total amount of dirt removed?

c) What does the slope of the dirt removal curve signify?

d) How can we find the total dirt removal at a specific time?

## Answers:

a) The derivative of the amount of dirt with respect to time represents the instantaneous rate of change or speed at which dirt is removed.

b) The integral over time provides the total amount of dirt removed up to a specific time.

c) The slope of the dirt removal curve at a particular time shows the instantaneous rate of change.

d) Total dirt removal at any time can be found by integrating the rate of removal over that period.

The question relates to the rate of change in the amount of dirt removed over time. We are looking for both the instantaneous and cumulative amount of dirt removed. Now, let's consider each of the provided cases.

**a) Derivative of dirt amount with respect to time:** The derivative of the dirt amount with respect to time can be represented as the instantaneous rate of change of dirt removal. It is like the velocity at a specific point of time. In calculus, the derivative at a point is the slope of the tangent at that point on the graph.

**b) Integral of dirt amount over time:** The integral of the dirt amount with respect to time gives the total amount of dirt removed up to a particular time. It's like the distance travelled over a certain time period. This integral value represents the cumulative or total dirt removed up to a given point in time.

**c) Slope of the dirt removal curve:** The slope of the dirt removal curve is the rate at which the amount of dirt is being removed, it is also the derivative of the dirt amount with respect to time at a specific point.

**d) Total dirt removal at a specific time:** The total dirt removal at a particular time can be found by integrating the rate of dirt removal from the start time to the given time.