8. Suppose an investment of $5,000 doubles every 12
years. How much is the investment worth after 36
years? After 48 years?
Write and solve an exponential equation
amount = 5000(2)^(t/12), where t is the number of years.
plug in 36, 48
actually, you should be able to do this in your head.
after 12 years ----> 10,000
after 24 years ----> 20,000
after 36 years ----> 40,000
after 48 years ----> 80,000
Ohhhh... Thank YOU SO MUCH Reiny
Well, it seems like you have a pretty amazing investment there. Doubling every 12 years? That's quite the money tree! Now, let's solve this puzzle together.
Let's assume X represents the initial investment of $5,000, and we want to find out how much it's worth after a certain number of years.
After 12 years, the investment doubles, so it becomes 2X. After another 12 years (24 years in total), it doubles again and becomes 4X. After yet another 12 years (36 years in total), it doubles once more and becomes 8X.
So after 36 years, your investment would be worth 8 times the initial amount, which is 8 * $5,000 = $40,000. That's a nice chunk of change!
Now, you also asked about the worth after 48 years. By following the same logic, after 48 years, the investment would double once more and become 16X. So, in this case, it would be 16 * $5,000 = $80,000.
Remember, these calculations all depend on the assumption that the investment continues to double every 12 years. It's important to note that actual investment scenarios may vary. But hey, who doesn't love a little imaginary money doubling every few years?
To solve this exponential equation, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A is the final amount
P is the principal amount (initial investment)
r is the annual interest rate (in decimal form)
t is the number of years
n is the number of compounding periods per year
In this case, the investment doubles every 12 years, which means the growth rate is 100% (or 1) with a compounding period of 1 year. Therefore, we can rewrite the equation as:
A = P(1 + 1/1)^(1*t)
A = P(2)^t
Now we can find the value of the investment after 36 years:
A = $5,000 * (2)^36
A ≈ $5,000 * 68.719476736
A ≈ $343,597.38
After 36 years, the investment is worth approximately $343,597.38.
Similarly, we can find the value of the investment after 48 years:
A = $5,000 * (2)^48
A ≈ $5,000 * 281.47497671
A ≈ $1,407,374.88
After 48 years, the investment is worth approximately $1,407,374.88.
To solve this problem, we need to use the formula for exponential growth:
A = P * (1 + r)^t
Where:
A = the final amount or value of the investment
P = the initial amount or value of the investment
r = the growth rate per period (expressed as a decimal)
t = the number of periods
In this case, the investment doubles every 12 years, so the growth rate per period would be 100% or 1. The initial amount (P) is $5,000.
Let's calculate the value after 36 years:
A = $5,000 * (1 + 1)^36
A = $5,000 * 2^3
A = $5,000 * 8
A = $40,000
Therefore, the investment will be worth $40,000 after 36 years.
Now let's calculate the value after 48 years:
A = $5,000 * (1 + 1)^48
A = $5,000 * 2^4
A = $5,000 * 16
A = $80,000
Therefore, the investment will be worth $80,000 after 48 years.