Please help me with this problem. In 1626,Peter Minuit of the Dutch West India Company purchased Manhattan Island from Native Americans for $24. Assuming an exponential rate of inflation of 6% per year,how much will Manhattan be worth in 2010?
To calculate the worth of Manhattan Island in 2010, we will use the formula for calculating exponential growth:
Future Value = Present Value * (1 + r)^n
Where:
Present Value = $24 (the purchase price in 1626)
r = 6% (the inflation rate per year, expressed as a decimal: 0.06)
n = 2010 - 1626 (the number of years from 1626 to 2010)
Let's calculate it step by step:
Step 1: Convert the inflation rate to a decimal:
r = 6% = 0.06
Step 2: Calculate the number of years:
n = 2010 - 1626 = 384
Step 3: Calculate the future value using the formula:
Future Value = $24 * (1 + 0.06)^384
Step 4: Calculate the result:
Future Value = $24 * (1.06)^384
Using a calculator, the result is approximately $2.27 billion.
Therefore, Manhattan Island would be worth approximately $2.27 billion in 2010, assuming an exponential inflation rate of 6% per year.
To calculate how much Manhattan would be worth in 2010, taking into account an exponential rate of inflation of 6% per year, we need to use the compound interest formula.
The formula for compound interest is: A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (initial value)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for
In this case, the initial amount (P) is $24, the annual interest rate (r) is 6% (or 0.06 as a decimal), the time period (t) is 2010 - 1626 = 384 years, and assuming it is compounded annually (n = 1).
Plugging in the values into the formula, we have:
A = 24(1 + 0.06/1)^(1 * 384)
Now, let's calculate this value:
Step 1: Calculate the value inside the parentheses
(1 + 0.06/1) = 1.06
Step 2: Raise the value inside the parentheses to the power of (1 * 384)
1.06^384 = 29264.45 (approximately)
Step 3: Multiply the principal amount by the calculated value
A = 24 * 29264.45 = $702,946.80 (approximately)
Therefore, Manhattan would be worth approximately $702,946.80 in 2010, assuming an exponential rate of inflation of 6% per year.