I did an experiment where I had to measure how fast water flowed out of a container at different times and with different volumes. I also had to record the values into a table and graph them. (I got an exponential function). I am to answer the following questions and I do not know how to answer them:
1. a) What connections can you make between instantaneous flow rates at a specific time and tangents to the graph?
b) Approximate the instantaneous rate of flow, when 750 mL of water has been collected in the measuring cup, by using your graph and a series of secants containing the point.
I don't understand what is meant by a series of secants.
2. a) Specifications for the water tower require that the rate of flow cannot be less than half the initial instantaneous flow rate. What is the initial instantaneous rate of flow (the rate of flow at the start of the experiment)?
would this be the average rate of water flow over the whole time interval?
secants connect two points on the graph. The slope of the secant is the average rate of change during that interval.
The tangents are just secants where the interval has zero length -- their slopes are the instantaneous rates of change.
The initial instantaneous rate is just the slope of the tangent at t=0.
1. a) When considering the instantaneous flow rates at a specific time, you can draw connections to the tangents of the graph by examining the slope. The tangent to a point on a graph represents the instantaneous rate of change or the rate at which the water is flowing at that precise moment. It shows how quickly the water is flowing at that particular time, and the slope of the tangent line represents the rate of flow at that instant.
b) To approximate the instantaneous rate of flow when 750 mL of water has been collected, you can use a series of secants. A secant line is a straight line that intersects a curve at two points. In this case, you can draw multiple secant lines on the graph that pass through the point where you have collected 750 mL of water and intersect the curve. By calculating the slope of each secant line, you can determine the average rate of flow between the two points on the graph. The more secant lines you draw, the closer you can approximate the instantaneous rate of flow when 750 mL has been collected.
2. No, the initial instantaneous rate of flow is not the average rate of water flow over the whole time interval. The initial instantaneous rate of flow refers to the rate of flow at the very beginning of the experiment, at the start. It is the rate at which the water is flowing when you first start measuring. To determine the initial instantaneous rate of flow, you need to examine the graph and identify the slope of the tangent line at the very beginning of the graph or experiment. The slope at the start represents the initial instantaneous rate of flow.