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MATH 101 — ASSIGNMENT 2
Learning objectives
• Use differential equations to understand exponential decay
• Use numerical integration to estimate error intervals from data
• Compute volumes of real-life objects
• Calculate distance travelled along a non-linear curve.
• Apply the concept of expectation
• Compute partial fractions for roots of degree > 1.
• Learn about a UBC alumni
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. Indicate members who have not contributed using the comment “(non-contributing)” .
Reflection questions
Reflection questions encourage you to think about how mathematics is done. This is an important ingredient of success. Reflection questions contribute to your engagement grade.
1. In written assignments you are asked to solve difficult problems as a team. Understanding how to approach or begin a problem is often the most challenging aspect of problem solving. Discuss the following prompts together and write a one or two paragraph response addressing all three.
(a) What strategies does your team use when you are stuck and do not know how to approach or begin to solve a problem?
(b) Which strategies have you found to be the most effective?
(c) Is the way you approach difficult problems in math similar to how you approach difficult problems in other subject areas? Explain why or why not.
2. Each member of your team should submit an individual response for part (a). Part (b) should be done as a group. You will not be evaluated on what you write here; use this question as a prompt for your own preparation.
(a) Make a realistic study plan for yourself for the final exam. You may want to consider: timelines (for your schedule), topics (in order of importance), resources, different types of preparation (review versus exam practice), and mental and physical health (sleep, movement breaks).
(b) Compose a brief message to the rest of the class to motivate them for the final exam. You may use words, drawings, memes, or any other medium of your choice (The best messages may be posted anonymously on Piazza!).
Assignment questions
The questions in this section contribute to your assignment grade. Stars indicate the difficulty of the questions, as described on Canvas.
First, some notes:
• In all your answers, when giving a result that is a numerical value, specify the unit when possible.
• Also, take a moment to review the difference between decimal places and significant digits.
• Some of the non-mathematical facts in this assignment are true, some are completely made up.
3. (10 marks) Radioactivity
The radioactive isotope wanowanium-18 is present in small quantities in many UBC students and alumni, and is constantly replenished until the student graduates. After this, the wanowanium-18 decays into stable wisium-22 at a rate proportional to the amount of wanowanium-18 present. Suppose that C(t) is the amount of wanowanium-18 present in a student at time t, and that the half-life of wanowanium-18 is 4.6 years. (The half life is the time taken for the amount of wanowanium-18 to decay to exactly one half of what it was originally.)
(a) (★☆☆☆) Find the general solution C(t) of the relevant differential equation below, as well as the particular solution for an initial amount C(0) = C0 :
dt/dC = -kC
(b) (★☆☆☆) Based on the half-life of wanowanium-18, find the value of the constant k and specify its unit.
(c) (★☆☆☆) From the differential equation, and the value and unit of k, give an interpretation for the meaning of this constant. Justify your answer.
(d) (★☆☆☆) In 2021, a team of scientists measured that Bjarni Tryggvason had 0.062% of the amount of wanowanium-18 found in fresh UBC graduates. When did Bjarni Tryggvason graduate according to this data?
4. (16 marks) Waste in space
The waste compartment of Bjarni Tryggvason’s space shuttle Discovery had a leak. The following measurements, made at the start of each day of the mission, show the rate at which waste was escaping (in kg/day) into space.
Time (days) 0 1 2 3 4 5 6
Rate r(t) 4 8 12 19 29 41 55
(a) (★☆☆☆ ) In the table, find an overestimate and an underestimate of the quantity of waste that escaped during the first day.
(b) (★☆☆☆ ) Make an overestimate and an underestimate of P, the total quantity of waste that escaped during the whole period for which we have data.
(c) (★☆☆☆ ) How often would measurements have to be made to find an overestimate and an under- estimate for P, which differ by exactly 1 kg from each other?
(d) (★☆☆☆ ) Enter the table values in a spreadsheet, plot the data points on a graph, and fit a quadratic function to these data points (also called quadratic regression). Show the plot you obtain, and write the equation of the fitted function in terms of t. Keep 4 significant digits for decimal numbers.
(e) (★★☆☆ ) We now assume that in reality, the rate r(t) followed a smooth quadratic function (such as the fitted one in the previous question). Using the trapezoidal rule and its error bound, find a tighter interval for the true value of P, than the one found in question (b). Give the results with 6 significant digits.
(f) (★☆☆☆ ) Does the Simpson approximation fall in the interval found in the previous question? Given the current assumption that r(t) follows a quadratic equation, can we find an even tighter interval for the true value of P?
(g) (★☆☆☆ ) We now change our assumption, and assume that in reality, r(t) follows the equation of a polynomial of degree 4. In your spreadsheet, change the fit function to a polynomial of degree 4. Show the plot you obtain, and write the equation of the fitted function in terms of t. Keep 4 significant digits for decimal numbers.
(h) (★★☆☆ ) Given our new assumption, can we find an even tighter interval for the true value of P?
5. (4 marks) Fixing the leak
On day 6 of the STS-85 mission, Bjarni Tryggvason identified the leak. It was due to a broken part in the waste compartment pipe system. Unfortunately, they didn’t have a spare for this part (how foolish!), so they had to 3D print it, with one of the first portable prototypes of selective laser melting. In order to gauge how much metal was needed to print the part, Bjarni Tryggvason needed its volume.
(★★☆☆ ) Calculate the volume of the solid obtained by rotating the region bounded by the curves y = 9 − 5x + x2 and y = −9 + 5x − x2 and the lines x = 1cm and x = 4cm, around the line y = 6cm. Don’t hesitate to use Desmos to help you visualize this.
6. (8 marks) Getting home
The leak was thankfully fixed very fast on day 7. However, on day 9 of the mission, Bjarni Tryggvason realized that they might not have enouh fuel to reach their destination. He knew the trajectory the shuttle had to take, but unfortunately the distance calculator started misbehaving and only returning non-sensical values. He therefore had to compute the length of the remaining trip, by hand.
Fortunately, the amount of wanowanium-18 within him was still high enough at that time for him to remember the following (it was testable material at the time):
If an arc or curve is described by the equation y = f(x), between the limits a ≤ x ≤ b, then we can calculate its length using the following formula:
(a) (★☆☆☆) Calculate the distance to destination Dd , which is the arc length of the curve y = 3/2(x − 1)3/2, from x = 1 million kilometers to x = 4 million kilometers.
(b) (★★☆☆) Bjarni Tryggvason knew the shuttle like no other, in particular how the engine worked, and how unpredictable it could be. Back on Earth, they had observed that the engine’s fuel consumption was random and independent from speed, but at least it followed a Beta distribution, whose probability distribution function is:
where α and β are distribution parameters, x ∈ [0, 1] is the engine’s fuel consumption in L/km and
B(α,β) is the Beta function, defined as 
An interesting property of the Beta function is that 
(great exercise to practice intergration by parts!)
Measurements on Earth showed that parameter values were α = 2 and β = 3 for the shuttle Discovery.
Under that distribution, what is the total amount of fuel expected to be consumed for the remainder of the trip? Write your answer in terms of Dd and the distribution parameters, and then compute its numerical value.
(c) (★☆☆☆) Knowing that the tanks still contained around 2000m3 of fuel at that time, did Bjarni Tryggvason and his crew reach their destination?
7. (7 marks) A “nice” reward
Once they all came back on Earth, NASA celebrated with the crew, and to reward Bjarni Tryggvason, they gave him a letter. As he opened the letter, he read:
“Congratulations Bjarni Tryggvason! You have made this mission a great success! As a reward, we offer you these two wonderful integrals, they are yours to solve.
All the best, NASA
P.S: Note that when a fraction has a root of degree n in the denominator, like 
it can be decomposed into a sum of terms with all the intermediate root degrees from 1 to n: 
where
the {Ai }1≤i≤n are constants.”
(a) (★☆☆☆) Solve the following integral: 
(b) (★★☆☆) Solve the following integral: 