GROUP PROJECT 3
Your group submission must be typed, and marks will be awarded for communication (see Canvas assign- ment).
Learning objectives:
• Be able to solve relates rates questions using the chain rule.
• Interpret derivatives in the context of a model.
• Apply mathematical concepts to models of physical processes.
• Practice the skill of using parameters rather than specific numbers in mathematical expressions.
• Be able to clearly and effectively communicate mathematical content in prose.
Contributors
On the first page of your submission, list the student numbers and full names (with the last name in bold) of all team members.
After submitting this assignment, you will provide feedback on your teammates’ contributions using iPeer. If there is zero contact with a group member, please mark NP (for ‘non-participating’) beside their name in this list, and award a 0 on iPeer.
Reflection question
Consider how you have been working as a team, both in small classes and on your written assignments.
In one paragraph, describe your greatest strength as a team and one aspect of teamwork that your team does especially well. In a second paragraph, describe one aspect of teamwork that your team can improve on and describe how you will work together to improve this aspect of teamwork in future small classes (or outside of small class) to help all your team members succeed in this course.
Assignment questions
1. An elastic tube is being stretched so that its diameter grows while its length stays the same. Initially, its inner radius is 1cm and the thickness of the material making up the tube is 1 cm.
The tube maintains a cylindrical shape as it stretches. The material making up the tube deforms in such a way that the thickness of the tube (i.e. the difference between the outer radius and the inner radius) is always inversely proportional to the inner diameter. That is, if the tube has thickness x(t) and inner diameter d(t) (both of which change with time, t), then
for some constant c.
At a certain time t0 , the inner diameter is growing at a rate of 2 centimetres per second, and the inner diameter is 3 cm.
(a) (2 points) How fast is the thickness of the tube changing at time t0 ?
(b) (3 points) Calculate the rate of change of the volume of the material making up the tube at time t0 if the length of the tube is ℓ centimetres. (Note: this is typeset in Latex as \ell.)
(c) Finally, assume that the tube has the mass 2 grams, which is constant. The density of the material making up the tube is found by dividing its mass by its volume.
i. (1 point) Based on the sign of your answer from (b), will the density be increasing or decreasing at time t0 ? Give an explanation that doesn’t involve actually calculating the derivative of the density.
ii. (2 points) Calculate the rate of change of the density of the material making up the tube at the time t0 .
2. Blood in the centre of a blood vessel flows faster than blood near the edges. Suppose the velocity v of blood is given by
v = 375(R2 − r2)
where R is the radius of the blood vessel and r is the distance of a layer of blood flow from the centre of the vessel. We’ll measure R and r in mm, and we’ll measure v in mm per minute.
(a) Imagine a bacterium pierces the wall of a blood vessel at time t = 0. (So, its distance from the centre of the vessel at that time would be R.) As it flows along with the blood, it wiggles its way straight through the centre of the vessel, and comes out the other side.
The bacterium moves through the cross-section of the vessel (i.e. the path shown above) at a constant rate of 1 mm/min. That is, if r is the bacterium’s distance from the centre of the vessel, |dt/dr| = 1 mm/min over the entire domain of dt/dr .
i. (1 point) Give an explicit expression for r in terms of time, for the duration of the bacterium’s journey, if R = 8 mm.
ii. (2 points) Which image below best displays the path the bacterium might take, and why?
iii. (2 points) Suppose again R = 8. How fast is the velocity of the blood experienced by the bacterium changing when it begins its journey, at r = 8? What about when it’s in the middle of its journey, at r = 0?
Note: be careful with your work here, since r(t) is not differentiable everywhere.
iv. (1 point) Would the bacterium experience a more rapid blood velocity when it was close to the edges of the vessel, or close to the centre?
v. (1 point) Would the bacterium experience a more rapid change in blood velocity when it was close to the edges of the vessel, or close to the centre?
(b) Cold weather is causing a blood vessel to contract at a constant rate, say dt/dR = −0.01 mm per minute. Still, the velocity of blood r millimetres from the centre is given by v = 375(R2 − r2).
i. (1 point) How fast is the velocity of the blood r mm from the centre changing when the radius of the vessel is 8 mm?
ii. (1 point) Is blood closer to the centre experiencing a faster or slower change in velocity than blood farther from the centre?