Math - Calculus?

Question

A bacteria culture starts with 1000 bacteria. Two hours later there are 1500 basteria. Finf an exponential model for the size of the culture as function of time t in hours, and use the model to predict how many bacteria there will be after two days.

Answer 1

Exponential functions mean the equation will look like f(t)=1000*k^t in order to create an exuation you need to find k

Put in the 1500 and the 2 hours: 1500=1000k^2
Solve for k: k^2=1.5 k=1.2247
f(t)=1000*1.2247^t

Put in 48 hours for 2 days: 1000*1.2247^48=16834112 bacteria after 2 days.

Answer 2

let time interval "unit" = 2 hours
k = 1.5 [this is the 'proportion of increase" for a "unit-time"]

S = starting number

f(t) = S * k^t

two days = 48 hours => t = 24

f(24) = s * k^24 = 1000 * (1.5)^24
f(24) = 1000 * 16834.112 = 16834112

so after 2 days there will be 16,834,112 bacteria

Answer 3

y = 1000 * sqrt(1.5)^t

This will explain the number after 2 hours. sqrt is of course the squared root and ^ the power. The total number after two days will be:

y = 1000*sqrt(1.5)^48 = 16.834.112

Answer 4

Let

t = time
N = number of bacteria at time t

N = 1000e^(kt)
1500 = 1000e^(2k)
e^(2k) = 1500/1000 = 3/2
ln{e^(2k)} = ln(3/2)
2k = ln(3/2)
k = (1/2)ln(3/2)

N = 1000e^(kt)
N = 1000e^{(1/2)(ln(3/2))t} = 1000e^{(t/2)(ln(3/2))}

After two days t = 48 hours

N = 1000e^{(48/2)(ln(3/2))} = 1000e^{(24)(ln(3/2))}
N = 16,834,112

Answer 5

N = ab^t
1500 = 1000b^2
b^2 = 1.5
b = 1.22474
N = 1,000(1.5)^t/2, t in hours
N = 1,000(1.5)^24
N = 16,834,112