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Patricia Glas... Efficenter Login $\mathrm{S}$. NCSA Client Recru. LogMein My Account - Pay. imported ec fogr b bg All B CENGAGE $\quad$ MINDTAP Search this course Ch 04: Assignment - Time Value of Money member for at least 14 years. Remember to change your calculator back to END mode immediately after you have calculated the correct answer. In 1626, Dutchman Peter Minuit purchased Manhattan Island from a local Native American tribe. Historians estimate that the price he paid for the island was about $\$ 24$ worth of goods, including beads, trinkets, cloth, kettles, and axe heads. Many people find it laughable that Manhattan Island would be sold for $\$ 24$, but you need to consider the future value (FV) of that price in more current times. If the $\$ 24$ purchase price could have been invested at a $5.75 \%$ annual interest rate, what is its value as of 2012 ( 386 years later)? $\$ 56,550,493,062.83$ $x$ $465,033,067,022.25$ $\$ 74,646,650,842.94$ $\$ 48,067,919,103,41$ Points: $0 / 1$ Explanation: Close Explanation $1626=386$ ), so $\mathrm{N}=386$. You are finding the FV of the $\$ 24$ fump sum, so there are no intermediate payments (PMT $=0$ ). The calculation is as follows: $\quad$ ow

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Clayton

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Solution By Steps

Step 1: Identify Variables

Principal (PV): $24

Interest Rate (I/Y): 5.75% per year

Number of Periods (N): 386 years

Future Value (FV): To be calculated

Step 2: Apply the Future Value Formula

The formula for future value is:

FV = PV imes (1 + r)^n

Where:

-

PV is the present value (initial amount)

-

r is the annual interest rate

-

n is the number of periods

Step 3: Substitute Values into the Formula

FV = 24 imes (1 + 0.0575)^{386}

Step 4: Calculate the Future Value

FV = 24 imes (1.0575)^{386}

Using a calculator to compute

(1.0575)^{386}:

(1.0575)^{386} \approx 1,939,720,544,284.38

Step 5: Multiply by the Principal

FV = 24 imes 1,939,720,544,284.38

FV \approx 46,553,293,062,825.12

Final Answer

The future value of $24 invested at 5.75% annual interest rate for 386 years is approximately $46,553,293,062,825.12.

This calculation demonstrates the power of compound interest over a long period, showing how a small initial investment can grow to an enormous sum given enough time and a reasonable interest rate.

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