Assume the national debt D increases at an annual rate of 5% and that the debt in 2000 was 57 billion dollars. This means that D = 57(1.05t ), where t is the number of years after 2000.
In what year will the debt have doubled?
that will be when 1.05^t = 2
Now add that t to 2000
To find the year when the national debt has doubled, we need to determine when D = 2D₀, where D₀ is the initial debt in the year 2000.
Let's start by substituting D₀ = 57 billion dollars into the equation D = 57(1.05t):
2D₀ = 57(1.05t)
Divide both sides by D₀:
2 = 1.05t
Now, we can solve for t by taking the natural logarithm of both sides:
ln(2) = ln(1.05t)
Using logarithmic properties, we can bring down the exponent t:
ln(2) = t ln(1.05)
Now, divide both sides by ln(1.05):
t = ln(2) / ln(1.05)
Using a calculator, we can evaluate the expression on the right side:
t ≈ 14.21
Since t represents the number of years after 2000, we round up to the nearest whole number:
t = 15
Therefore, the debt will have doubled in the year 2000 + 15 = 2015.