Answer:Since time cannot be negative, we can conclude that it would take approximately 3.02 years (rounded to 2 decimal places) for the capital of $500000 to accumulate to the sum of $86400 at the interest rate that capitalizes 20% per year.
Step-by-step explanation:
To calculate the time it takes for a capital of $500000 to accumulate to the sum of $86400 at an interest rate that capitalizes 20% per year, we can use the compound interest formula:A = P * (1 + r/n)^(n*t)where:
A = the final amount (the sum of $86400)
P = the principal amount (the initial capital of $500000)
r = the annual interest rate in decimal form (20% = 0.20)
n = the number of times interest is compounded per year (assuming annually, so n = 1)
t = the time in years (unknown, what we want to find)Substitute the known values into the formula:86400 = 500000 * (1 + 0.20/1)^(1*t)Simplify:86400 = 500000 * (1.20)^tTo isolate 't', divide both sides by 500000:0.1728 = 1.20^tNow, take the logarithm (base 10 or natural logarithm, doesn't matter) of both sides to solve for 't':log(0.1728) = log(1.20^t)t * log(1.20) = log(0.1728)t = log(0.1728) / log(1.20)Using a calculator:t ≈ -3.02