If an account that earns interest compounded continuously takes 37 years to double in value, how long will it take to triple in value?
Here's what I think...I divided 37 with 2 and multiplied that value with 3 and so my answer is 55.5 years. Am I correct? If not, please help me get the right answer! Thanks in advance.
continuous growth is given by the exponential equation
y = a(e)^kt
let a=1 then for doubling y = 2
2 = e^37k
37k = ln 2
k = ln 2/37
so y = a(e)^(ln2/37)t
so now you want y to be 3
3 = e^(ln2/37)t
(ln2/37)t = ln3
t = 37ln3/ln2 = 58.6 years
You titled your post "exponential" but totally ignored that important property.
You treated the relationship as "linear"
thanks! this tells me i need to review the chapter...
921
To solve this problem, you need to use the formula for continuous compound interest: A = P * e^(rt), where A is the final amount, P is the principal (initial amount), r is the interest rate, and t is the time in years.
In this case, since the account takes 37 years to double in value, we can set up the equation:
2P = P * e^(r * 37)
Next, we will divide both sides of the equation by P to isolate e^(r * 37):
2 = e^(r * 37)
To solve for r, take the natural logarithm of both sides:
ln(2) = r * 37
Now, divide both sides by 37 to solve for r:
r = ln(2) / 37
With this value for r, we can now determine how long it will take for the account to triple in value. Let's set up the equation again:
3P = P * e^(r * t)
Divide both sides by P:
3 = e^(r * t)
To solve for t, take the natural logarithm of both sides:
ln(3) = r * t
Finally, divide both sides by r to solve for t:
t = ln(3) / r
Using the value of r we calculated previously, you can plug it into this equation to find the answer:
t = ln(3) / (ln(2) / 37)
Calculating this expression, the approximate value for t is about 55.934 years, or rounded to one decimal place, 55.9 years.
Therefore, the correct answer is that it will take approximately 55.9 years for the account to triple in value, not 55.5 years as you initially thought.