Rate of change question: "Express t in terms of N"

[Monkeyseat]

Hi,

Question:

Observations were made of the number of bacteria in a certain specimen. The number N present after t minutes is modelled by the formula N = A(c^t) where A and c are constants. Initially there are 1000 bacteria in the specimen.

a) Write down the value of A.
b) Given that there are 12,000 bacteria after 60 minutes, show that the value of c is 1.0423 to four decimal places.
ci) Express t in terms of N.
ii) Calculate, to the nearest minute, the time taken for the number of bacteria to increase from one thousand to one million.

Working:

a)

When t = 0, N = 1000 so using N = A(c^t):

1000 = A(c^0)
A = 1000

b)

When t = 60, N = 12,000 so:

12,000 = 1000(c^60)
c^60 = 12
ln (c^60) = ln 12
60 ln c = ln 12
ln c = (1/60) ln 12
c = 12^(1/60)
c = 1.0423

ci)

This is the part that I can't get right.

N = A(c^t) and I know that A = 1000 and c = 1.0423. Therefore:

N = 1000(1.0423^t)
N/1000 = 1.0423^t
ln (1.0423^t) = ln (N/1000)
t ln 1.0423 = ln (N/1000)
t = (ln(N/1000))/(ln 1.0423)

The book says that the answer for this part of the question is t = (1/1.0423) ln (N/1000).

Can someone show me where I am going wrong? I will be able to do part (cii) after I figure out (ci).

Many thanks.

 

[stapel]

Observations were made of the number of bacteria in a certain specimen. The number N present after t minutes is modelled by the formula N = A(c^t) where A and c are constants. Initially there are 1000 bacteria in the specimen.

a) Write down the value of A.

Yes, A = 1000.

b) Given that there are 12,000 bacteria after 60 minutes, show that the value of c is 1.0423 to four decimal places.

Your work for this looks fine. :wink:

c-i) Express t in terms of N.

N = Ac^t

ln(N) = ln(Ac^t) = ln(A) + ln(c^t) = ln(A) + t*ln(c)

ln(N) - ln(A) = t*ln(c)

ln(N/A) = t*ln(c)

[ln(N/A)] / ln(c) = t

So I get the same answer you do. I would suspect the book's answer is a typo.

 

[Deleted member 4993]

Hi,

Question:

Observations were made of the number of bacteria in a certain specimen. The number N present after t minutes is modelled by the formula N = A(c^t) where A and c are constants. Initially there are 1000 bacteria in the specimen.

a) Write down the value of A.
b) Given that there are 12,000 bacteria after 60 minutes, show that the value of c is 1.0423 to four decimal places.
ci) Express t in terms of N.
ii) Calculate, to the nearest minute, the time taken for the number of bacteria to increase from one thousand to one million.

Working:

a)

When t = 0, N = 1000 so using N = A(c^t):

1000 = A(c^0)
A = 1000

b)

When t = 60, N = 12,000 so:

12,000 = 1000(c^60)
c^60 = 12
ln (c^60) = ln 12
60 ln c = ln 12
ln c = (1/60) ln 12
c = 12^(1/60)
c = 1.0423

ci)

This is the part that I can't get right.

N = A(c^t) and I know that A = 1000 and c = 1.0423. Therefore:

N = 1000(1.0423^t)
N/1000 = 1.0423^t
ln (1.0423^t) = ln (N/1000)
t ln 1.0423 = ln (N/1000)
t = (ln(N/1000))/(ln 1.0423)

The book says that the answer for this part of the question is t = (1/1.0423) ln (N/1000).

Can someone show me where I am going wrong? I will be able to do part (cii) after I figure out (ci).

Many thanks.

As far as I can see (I didn't put pencil to paper) - your calculations are correct.

Put t=60 - in their equation - and see what do you get for 'N'. Do the same for your equation. Then march on....

 

[Monkeyseat]

Thanks for replying both of you. Subhotosh Khan, when I substitute in t = 60 I get around 12,011 with my answer and with the book's answer I get 1.45 x 10^30. So I suppose that this suggests that my answer is probably correct.

However, this raises a problem when I come to do part (cii). What should I do here?

cii)

Using the book's answer

t = (1/1.0423) ln (N/1000)

When N = 1,000,000:

t = (1/1.0423) ln (1000000/1000)
t = (1/1.0423) ln 1000
t = 6.63

So it takes around 7 minutes (the book says that this is the 'correct' answer).

Using my answer

t = (ln(N/1000))/(ln 1.0423)

When N = 1,000,000:

t = (ln(1000000/1000))/(ln 1.0423)
t = (ln 1000)/(ln 1.0423)
t = 166.73

So it takes around 167 minutes.

I don't know which answer sounds more realistic. The book says the answer is 7 minutes, but they may have just got that from using their incorrect expression. I would've thought though that if it was just a typo when writing up the answers they would have got the second part correct.

Should I go with my answer of 167 minutes?

Thanks.

 

[BigGlenntheHeavy]

N(t) = Ac^t
N(0) = A = 1,000
N(t) = 1,000c^t
N(60) = 1,000c^60 = 12,000
c = 12^(1/60)
N(t) = 1,000[12^(t/60)]
1,000,000 = 1,000[12^(t/60)]
12^(t/60) = 1,000
t = 166.793
Check: N(t) = 1000[12^(166.793/60)] = 999,995.301....

 

[Deleted member 4993]

Since for N = 12,000 you had t = 60

for N = 1,000,000 you must have t > 60.

 

[Monkeyseat]

That makes sense, thanks.