Poly Year 2(Differential Equations) Problem Sum

Wireless- SG

8) A certain radioactive material decays at a rate proportional to the amount present.

Initially there is 50 mg of the material present. It is observed that the material has lost 10% of its original mass after 2 h. Find

(a) an expression for the mass of the material remainin at any time t, and

(b) the mass of the material after 4h

Ans 40.5 mg

PS: Only learnt seperable equation

Wireless- SG

9) A certain bacteria grow at a rate proportional to the number present. If the number of bacteria doubles in 4h, how many may be expected at teh end of 12h?

Ans 8 times

Again same type of qn.......

Terminator Hitman SG

those are easy questions...when u read proportional......it already tells u the formula...all it needs is to put in all data n u get the answers.

Terminator Hitman SG

try reading yer lecture notes again .there are 2 formulas, one for growth the other for decay...just use the right one n put in data n ya have d answer!

wee_ws SG

Hi,

Let x represent the amount of radioactive material (in mg) at time t (in hr).

Now dx/dt represents rate of change of x.

Given that rate is proportional to amount present, we use O-level knowledge (i.e., ratio, rate and proportion) to obtain

dx/dt = kx, where k < 0, since it is a decay (i.e. x decreases as t increases)

From here, we will apply the method of variable separable to find the general solution.

Since we are given some other information:

x = 50 when t = 0,

x = 0.9(50) when t = 2,

we will need to find the particular solution.

Thanks.

Cheers,
Wen Shih

bladez87 SG

wow what course is this sia.

Wireless- SG

wee_ws thanks again =]

got it

left got 10*, 11* and 12 * all star qns. Trying!

-Wanderer- SG

9) A certain bacteria grow at a rate proportional to the number present. If the number of bacteria doubles in 4h, how many may be expected at teh end of 12h?

Ans 8 times

Again same type of qn.......

This question is fairly simple. Don't need to read too much into it just to answer this question:

Let the initial amount of bacteria be x.

x doubles to 2x in 4 hrs.

2x doubles to 4x in 8 hrs.

4x doubles to 8x in 12 hrs.

Hence it is 8 times.

Simply put, x => 2x => 4x => 8x (with each arrow representing every 4 hrs that have passed)

I'm solving this in a very layman and simplistic way. So this may not be the proper working that your tutors might desire.

Wireless- SG

Ok the oni 1 left i dunno is this

11*) If the interest is compounded continuosly at a constant rate of 4% per annum, how long will it take a bank deposit tto double?

Let y be the amt of money in the bank at any time t (in years)

dy/dt=0.04y

(25/y)dy = 1dt

integrate

25ln|y|=t+K

y=e^([t/25]+K)

Then i dunno how to do

bladez87 SG

i donno if this works. but there is a formula that financial advisors use to calculate how long it takes to double a principle amount.

72/interest rate

=72/4 = 18years.

Wireless- SG

close 17.3 years is the ans.

Nid differential eqns working =/

Wireless- SG

Nvm finally got it. Should find the original amt 1st