In a country where inflation is a concern, prices have risen by 50 % over a 3-year period.
(a) By what percent B do the prices rise each year?
Find the time t it takes for prices to rise by 8%
Enter your answers to two decimal places.
b = ____ %
t = _______ years
(b) Continuous Growth rate to model inflation by function of the form P= Poe^kt ( o is not zero, it is p naught)
Enter your answer in four decimal places
k = ___%
1.5 = (1+r)^3
log 1.5 = 3 log (1+r)
1 + r = 1.1447
r = .145
so 14.5 percent
1.08 = 1.1447^n
find n
continuous
1.5 = e^3k
ln .5 = 3 k
(a) To find the percent by which prices rise each year, we can divide the total increase in prices over 3 years by the number of years.
The total increase in prices over 3 years is 50%.
Therefore, the percent increase each year is:
B = 50% / 3 years = 16.67% (rounded to two decimal places)
So, b = 16.67%.
To find the time t it takes for prices to rise by 8%, we can set up an equation:
8% = b * t
Rearranging the equation to solve for t, we have:
t = 8% / b
Plugging in the value of b, we have:
t = 8% / 16.67%
t ≈ 0.48 years (rounded to two decimal places)
So, t = 0.48 years.
(b) To model inflation using continuous growth, we can use the formula:
P = Poe^kt
Here, P represents the final price, Po represents the initial price, e is the base of the natural logarithm, k is the continuous growth rate, and t is the time.
In this case, we want to find the continuous growth rate k.
Since the function is in terms of percent, we need to convert the growth rate into a decimal.
If we let the initial price Po be 1 (for simplicity), and the final price P be (1 + 50%) = 1.50, we can substitute these values into the equation:
1.50 = 1e^(k*3)
Dividing both sides of the equation by 1, we have:
1.50 / 1 = e^(k*3)
Taking the natural logarithm of both sides, we get:
ln(1.50) = ln(e^(k*3))
Using the property of logarithms, ln(e^x) = x, the equation simplifies to:
ln(1.50) = k * 3
Now, solve for k by dividing both sides of the equation by 3:
k = ln(1.50) / 3
Calculating this value using a calculator or software tool, we get:
k ≈ 0.1547 (rounded to four decimal places)
So, k = 0.1547%, or approximately 0.1547%.