Final answer:
Caleb would need to invest approximately $1415.50 to reach $1900 in 5 years with daily compounding at a 6% rate. The closest larger amount from the provided options is $1500.
Step-by-step explanation:
To calculate how much Caleb would need to invest to reach a target amount due to compound interest, we can use the compound interest formula. The formula is A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial sum of money), r is the annual interest rate (decimal), n is the number of times that interest is compounded per year, and t is the time the money is invested for in years.
To solve for P, we rearrange the formula to P = A / (1 + r/n)^(nt). Given that A is $1900, r is 6% or 0.06 as a decimal, n is 365 (since the interest is compounded daily), and t is 5 years, we substitute these values in to find the initial investment required.
Using the given values:
- A = $1900
- r = 0.06
- n = 365
- t = 5
P = 1900 / (1 + 0.06/365)^(365*5)
Calculating this, we find that Caleb would need to invest approximately $1415.50 to reach $1900 in 5 years. However, since this option is not available in the choices (a) $1500 (b) $1600 (c) $1700 (d) $1800, and the investment amount required is less than all the provided options, the closest larger amount he could invest from the options provided would be $1500.