My question is how do I know that
N increases by a factor of 2.9 each day. (see below)
Let N(t) be number of bacteria after t days. N(t) = Pa^t for some constants P and a. N(2) = 5400 and N(6)=382,000.
a)before working problem, estimate value of N(5).
b) write down 2 equations for P and a, one when t=2 and one when t=6, and use these two equations to compute a and P. Round to three significant digits.
algebra - please help - drwls, Sunday, November 6, 2011 at 6:41am
(a) N increases by a factor of 2.9 each day. Divide 382,000 by 2.9 for N(5)
(b) 382,000 P*a^6
5400 = P*a^2
70.74 - a^4
a = (70.74)^(1/4) = ___
Once you have a, solve for P.
Because, as it shows above a = (70.74)^(1/4)= 2.9
and P*a^n multiples by a each day.
To answer part (a) of the question, we are given that N increases by a factor of 2.9 each day. From the information given, we know N(6) = 382,000. To estimate the value of N(5), we divide N(6) by the factor of increase:
N(5) = N(6) / 2.9 = 382,000 / 2.9 ≈ 131,724
Therefore, the estimated value of N(5) is approximately 131,724.
Now, let's move on to part (b) of the question. We are given two equations for N(t). The equation N(t) = Pa^t represents the number of bacteria after t days.
From N(2) = 5400, we can substitute t = 2 into the equation to get:
5400 = Pa^2
Similarly, from N(6) = 382,000, we can substitute t = 6 into the equation to get:
382,000 = Pa^6
Now, we have two equations:
5400 = Pa^2 ...(1)
382,000 = Pa^6 ...(2)
To find the values of P and a, we can solve these two equations simultaneously.
First, we can solve equation (1) for P:
P = 5400 / a^2
Next, substitute this value of P into equation (2):
382,000 = (5400 / a^2) * a^6
Simplify the equation:
382,000 = 5400 * a^4
Divide both sides of the equation by 5400:
70.74 = a^4
Take the fourth root of both sides:
a = (70.74)^(1/4)
Using a calculator, evaluate this expression to find:
a ≈ 2.963
Now that we have the value of a, we can substitute it back into equation (1) to find P:
5400 = P * (2.963)^2
Simplify the equation:
5400 = 8.794 * P
Divide both sides of the equation by 8.794:
P ≈ 614.733
Therefore, the values of P and a, rounded to three significant digits, are approximately P ≈ 614.733 and a ≈ 2.963.