A recent study shows that the number of Australian homes with a computer doubles every 8 months. Assuming that the number is increasing continuously, at approximately what monthly rate must the number of Australian computer owners be increasing for this to be true?

n = N e^kt

2 = 1 e^(k*8)
ln 2 = 8 k = .693
k = .0866

dn/dt = kN e^kt
at t = 0
dn/dt = k N = .0866 N
You must know how many there are now, N, to say how many per month are added.

To find the monthly rate at which the number of Australian computer owners must be increasing, we can use the formula for continuous compound interest. In this case, the formula can represent the growth rate of the number of computer owners over time.

The formula for continuous compound interest is:

A = P * e^(rt)

Where:
A = the final amount (number of computer owners)
P = the initial amount (number of computer owners)
e = the mathematical constant approximately equal to 2.71828
r = the rate of growth (monthly rate)
t = time in months

In this case, the initial amount is P, and after 8 months, the final amount A is double the initial amount.

So, let's say the initial amount is P. After 8 months, the final amount is 2P.

We can substitute these values into the formula:

2P = P * e^(r*8)

Simplifying the equation:

2 = e^(8r)

To approximate the monthly growth rate (r), we need to isolate the variable:

ln(2) = 8r

Now, we can solve for r:

r = ln(2)/8

Using a calculator, we can find the approximate value of r:

r ≈ 0.0866

Therefore, the monthly rate at which the number of Australian computer owners must be increasing is approximately 0.0866, or 8.66%.

To determine the monthly rate at which the number of Australian computer owners must be increasing for the number of Australian homes with a computer to double every 8 months, we can use the concept of exponential growth.

Let's denote the initial number of Australian homes with a computer as "N0" and the monthly rate of increase as "r". After 8 months, the number of homes would be doubled, so the final number of Australian homes with a computer would be 2 times the initial number, i.e., 2N0.

We can use the formula for exponential growth:

Final number = Initial number × (1 + rate)^time

Substituting the values:

2N0 = N0 × (1 + r)^8

Dividing both sides by N0:

2 = (1 + r)^8

To find the monthly rate at which the number of Australian computer owners must be increasing, we need to solve for "r". Taking the 8th root of both sides:

∛2 = 1 + r

∛2 - 1 = r

Now we can calculate the approximate monthly rate at which the number of Australian computer owners must be increasing:

r ≈ ∛2 - 1

Using a calculator, we can find that ∛2 is approximately 1.091. Subtracting 1:

r ≈ 1.091 - 1

r ≈ 0.091

Therefore, the monthly rate at which the number of Australian computer owners must be increasing for the number of Australian homes with a computer to double every 8 months is approximately 0.091 or 9.1%.