help with math word problems?

Question

2 questions

The population of a large herd of 6000 buffalo increases at a rate of 20% (.20) each year. In how many years will the population be nearly triple its present size?
a) 3 b) 4 c) 5 d) 6 e) 7

There are 2 specimens of a certain strain of bacteria in a special culture. These bacteria are known to triple in number ever 3 hours. How many will there be in the end of 24 hours?
a) 144 b) 216 c) 768 d) 13,122 e) 19,683

i dont know how to solve so plz explain or show work

Answer 1

d and d

Answer 2

to understand what is going on think of it this way:

the 6000 buffalo is what you start with, at the end of the first year you have 20% more or 6000x20%= 1200 more buffalo for a total of 6000+1200= 7200 buffalo this could have been done by multiplying 6000 by 1.2 (100%+20%) to get the answer directly. now do this again for the second year, third year etc until you get an answer nearest to 18000 this is a compounding type problem.

I got 17915.9 in the 6th year and 21499 in the 7th.
the formula can be written this way:
number of buffalo after x years = start number * (1.2)^x
so
18000 = 6000 * (1.2)^x
3=(1.2)^x
ln3 = x ln 1.2
x= (ln3)/ln(1.2) = 6.025 years

answer D fits best.

Part 2

if you start out with x bacteria and they triple (3x) every three hours the you have 3x bacteria at the end of 3 hrs. After 6 hours you have triple 3x or 9x bacteria so the formula becomes:

3^n is the number you have, where n= number of 3 hour blocks in the time given.
so 24/3= 8 three hour blocks, set n=8 and your answer is 3^8 or 6561 times the original number per specimen, since you have 2 specimens then
6561x2 gives you 13122 bacteria answer D again

Answer 3

I got answer D for both questions
multiply your 6000 buffalo by 20% and add it to the total
take that new total of buffalo and repeat the process of multiplying by 20% and adding in .. count how many times you multiply and add til you reach nearly 18000... which should be 6 times giving you 17950 ish...

Same theory.. you need to multiply 8 times to calculate for the 24 hours.. Multiply 2 x 3 to account for the 1st 3 hrs then multiply that answer (6) by three for the next 3 hours and so on until you have multiplied 8 times.. you should get 13122

Just trying to help in layman's terms.. :)

Answer 4

1. P = 6000 * (1.2)^y
Where y is the number of years.
So we want P = 18000
18000 = 6000 * (1.2)^y
3 = 1.2^y
y = log base 1.2 of 3
y = log 3 / log 1.2 = 6.02569 years

It depends on what they mean by nearly triple... I guess in 6 years it will be pretty close to triple, but in 7 years it will be more than triple.

2. In 24 hours the population will triple 8 times, so there will be 2*(3)^8 = 13122 after 24 hours.

Answer 5

I = P(1 + r)^t

I = final amount
P = intial amout
r = rate
t = time

I = 6,000 (1 + .2)^t
I = 6,000 (1.2)^t

6,000 x 3 = 18,000

18,000 = 6,000 (1.2)^t
3 = (1.2)^t
t = log3 / log1.2
t = 6.03 years or D


I = 2(3)^(t/3)

I = 2 (3)^(24/3)

I = 13,122 or D