The number of building permits in Pasco t years after 1992 roughly followed the equation n(t)=400e^.143t.what is the doubling time?
To find the doubling time, we need to determine the time it takes for the number of building permits to double from its initial value.
The general formula for exponential growth is given by the equation: N(t) = N(0) * e^(kt), where:
- N(t) represents the quantity at time t
- N(0) represents the initial quantity
- e is the base of the natural logarithm (approximately 2.71828)
- k is the constant rate of growth or decay
- t represents the time elapsed
In this case, the equation for the number of building permits is given as:
n(t) = 400 * e^(0.143t)
To find the doubling time, we need to solve for t when n(t) is twice the initial quantity, which is 400.
2 * 400 = 400 * e^(0.143t)
Dividing both sides of the equation by 400:
2 = e^(0.143t)
To isolate the exponent, we take the natural logarithm of both sides:
ln(2) = ln(e^(0.143t))
Using the logarithmic property ln(e^x) = x, we simplify:
ln(2) = 0.143t
Now, we can solve for t:
t = ln(2) / 0.143
Using a calculator, we find:
t ≈ 4.837 years
Therefore, the doubling time for the number of building permits in Pasco is approximately 4.837 years.
that would be where
e^.143t = 2
.143t = ln2
t = ln2/.143