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MAT 136 H1S ASSIGNMENT 3 (CORRECTED)

2022

This assignment consists of two parts: a written component given below, and a WeBWorK component. It is recommended that attempt at least some of the WeBWorK problems before the written problems. Your answers to the written component must be structured in full sentences, and you must explain your reasoning for each step of your computations.

1. Some chemists are attempting to quantify the number of fluorescent molecules in a cylindrical column which is 2 metres tall and has radius 5cm.1  Using spectroscopic techniques, they determine that the concentration of fluorescent molecules varies with distance from the bottom of the column according to the following graph:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0                                                   1                                                2

Distance from bottom of column (m)

Here the concentration refers to the number of molecules per unit volume, in units of molL1 . (One mole of molecules is approximately 6.02 · 1023  molecules.)

Derive an expression for the total number of fluorescent molecules in the column (in mol), and estimate its value.

2. A student has proposed that the height h of a tree (in metres) as a function of time t (in days) satisfies the differential equation

dh      V (t)

dt         h

where V (t) is how much the tree has been watered (in litres) in the 48 hours prior to t.

(a) If at time t the tree is watered at the rate of f(t) = 1 − cos2πt litres per day, compute V (t).      (b) Assuming the tree is watered as in the previous part, solve the differential equation (1) given that

at time 0 the tree is 2 metres tall.

(c) Give a reason why the above model is physically implausible.

3.   (a) Sketch the slope field for the differential equation y\ = xy(x + y) at the 25 points (x,y) where x and y are integers and −2 ≤ x,y ≤ 2. Plot the solution that goes through the point (1, 1).

(b) Use Euler’s method with step size 0.2 and starting at P = (1, 1) to estimate y(0), where y is the

solution of the differential equation going through P .

(c) Is the estimate obtained in (b) an overestimate or an underestimate? Justify your answer.