The phenomenon of a half-life can be expressed mathematically by the formula

"m=A exp(-kt)"
where m is the mass, t is the time, and A and k are positive constants. exp(-kt) refer to e^(-kt), while the minus sign refers to the fact that the substance is decreasing in mass.

Since the initial (t=0) mass is 3mg, m = 3 (using units of mg). Meanwhile, the half-life, t_1/2 is related to k by the formula
t_1/2 = ln(2)/k
k = ln(2)/t_1/2
Be sure to be consistent with your units of time. Since t is in years, t_1/2 = 10/12 and not 10.

Substitute the values in and you will get the expression
m = 3 exp(-0.832t)

For part two, simply substitute m = 0.046875 and solve for t.

P.S. I apologize for the rather tough to read math formatting. The message board doesn’t support certain codes that make math formatting much nicer. If things are hard to understand, do tell me and I might do a blog post on it.

P.P.S. My solution simply gave the formula to use, which doesn’t aid understanding. While we learn about the exponential in mathematics, the concept of the half-life isn’t emphasized enough in our school syllabus. There is also limited scope on this topic in chemistry. I believe the relevant formulae are introduced only in the physics syllabus. Hopefully reading through the relevant topics will help understand the phenomenon and the corresponding mathematical formulation.

KiausuLormee:

Can someone help in this questions:

1. Calculate the number of sides of a regular polygon if each interior angle is 170 degree.Proof 2 of the methods
2. Calculate the number of sides of a regular polygon if each exterior angle is 36 degree.

The two relevant formulae for this question are:

Angle sum of n-sided polygon = (n-2)*180 degrees
Sum of exterior angles of any polygon = 360 degrees

(Note: as important as knowing the formulae is knowing a bit about their significance and why they exist. For example, we can think of random triangles, squares or quadrilaterals and see how the formulae work for any shape we can draw.)

Moreover, a regular polygon is one where every side is the same length. Consequently, every angle is the same.

See if these hints help (it's good to try questions after some hints and see if we can progress further with them). The answers \"hidden\" below.

Answer 1 (Highlight to reveal):

Let the number of sides of the polygon be n. There are n interior angles.
Since each interior angle is 170 degrees, the angle sum is 170n.
Applying formula 1,
180(n-2) = 170n
10n = 360
n = 36
Hence there are 36 sides
** The second method is to consider the exterior angles and use equation 2. I've leave this for practice. Do feel free to ask again if you're not sure how to get it.

Answer 2 (Highlight to reveal):

Let the number of sides of the polygon be n. There are n exterior angles.
Since each exterior angle is 36 degrees,
36n = 360 (using formula 2)
n = 10
Hence there are 10 sides

One of the major downside of the TI-84 plus and TI-84 plus Silver edition is that they are bulky and inconvenient to bring around.


Thus I recommend the Pocket for the form factor.

That said, all 3 calculators essentially perform similarly. The silver edition and the pocket are supposedly slightly faster but if we key everything in correctly the speed difference is negligible in the context of a level type calculations.

archie2:

can someone help with this question

Q>> a bag contains 200 marbles. there are 60 red, 60 blue, 60 green and the remaining 20 consists of some yellow and white. If marbles are chosen from the bag without looking, what is the smallest number of marbles one must pick in order to ensure that, among the chosen marbles, at least 20 are of the same colour?

To answer this question, we must understand that we want to be SURE that, even without looking, we will get at least 20 or the same colour. To be sure, we must account for if we are unlucky; in fact, we want to consider the unluckiest case possible to get the answer. So, assuming we have the worst luck ever, which marbles will we be picking up such that we don't get at least 20 of the same colour?

Hint 1 (Highlight to reveal):

We might have picked the yellow and white ones, which will not give us 20 of the same. Thus, the answer is bigger than 20 because we could have picked those up and not get what we want.

Hint 2 (Highlight to reveal):

If we are really unlucky, we will not get 20 red/blue/green quickly. We could have picked 19 red and then start drawing blue and greens and not get what we want. Thus, the answer is bigger than 20 (from hint 1) + 19. Try to continue from this way of thinking and see if you can arrive at the answer such that we surely will get 20 of a kind.

Answer (Hightlight to reveal):

78.
We could have picked 20 white/yellow, 19 reds, 19 blues and 19 greens and STILL not satisfy the criteria. After that, the next marble we pick would surely given us 20 red/green/blue marbles. Hence 20+19+19+19+1=78.

If he has been doing relatively well so far then there should be nothing to worry about. (Granted there has only been a CA1 test to base this on. The mid-terms, taken together with other class tests/quizzes should give a better indication of how he is doing.)


Workload for classwork typically only pick up later in the year and in his upper secondary years so it will not be uncommon for some free time. If he is doing well then that is definitely a good sign because he’s got more to spare if things start to get more difficult.