Microsoft Word - MTH 251_lab_manual_cover.doc 45 178 Problem 24.3 Three containe
ID: 2890625 • Letter: M
Question
Microsoft Word - MTH 251_lab_manual_cover.doc 45 178 Problem 24.3 Three containers are shown in figures 24.5-24.7. Each of the following questions are in reference to these containers. Suppose that water is being poured into each of the containers at a constant rate. Let hs, h,, and h, be the heights (measured in cm) of the liquid in containers 24.5-24.7, respectively, seconds after the water began to fill the containers. What would you expect the sign to be on the second derivative functions h,, h", h," while the containers are being filled? (Hint: Think about the shape of the curves y = hs (1) . y-h(t), and y=h(1).) 24.3.1 24.3.2 Suppose that water is being drained from each of the containers at a constant rate. Let , and h, be the heights (measured in cm) of the liquid remaining in the containers seconds after the water began to drain. What would you expect the sign to be on the second derivative functions h,". h," while the containers are being drained?Explanation / Answer
Look in terms of how height is responding to the change in the cross sectional area which being the driving factor here. The flow rate into the containers is constant that means same amount (volume) of water is flowing into each of them.
24.3.1
In Figure 24.5, the cross sectional area is constant, hence the height of water will be increasing or decreasing with a constant rate which means that second derivative of h(t) will be zero.
In Figure 24.6, the cross sectional area is increasing such that more and more water will be required to fill the same depth, but since the flow rate is constant, the same amount of water will fill less and less height as water fills up the container. This means that the rate of change of height is decreasing as water level goes up. That is the second derivative of h(t) will be negative.
You can argue the last figure in a similar fashion. As we move up, water will fill more and more height such that the rate with which height is increasing is also increasing. That is the second derivative of h(t) will be positive.
24.3.2
This is simply an inverse problem of the previous part. Figure 24.6 and 24.7 will exchange their roles. I am sure you can explain in a similar manner.
Figure 24.5 ---> second derivative will be zero.
Figure 24.5 ---> second derivative will be positive (now water level moves down i.e. in the direction of decrease in cross sectional area).
Figure 24.5 ---> second derivative will be negative (now water level moves down i.e. in the direction of increase in cross sectional area).
Hope this helps :)