Final answer:
To determine the time for $1800 to grow to $3300 at a 7% interest rate compounded monthly, one must use the compound interest formula and solve for the time variable t. This process involves dividing both sides of the equation by the initial amount, taking the natural logarithm, and then solving for t which will give the number of years rounded to the nearest hundredth.
Step-by-step explanation:
Compound Interest Calculation
The question pertains to the calculation of time required for an investment to grow from $1800 to $3300 at a 7% annual interest rate compounded monthly. This involves the formula for compound interest, which 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 amount 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 or borrowed for, in years.
In this scenario, we need to solve for the variable t:
A = $3300 (final amount)
P = $1800 (initial amount)
r = 0.07 (7% annual interest rate, as a decimal)
n = 12 (since the interest is compounded monthly)
t = ? (the time to solve for)
The formula becomes: $3300 = $1800(1 + 0.07/12)12t.
To find t, we need to isolate it on one side of the equation. The steps to do this include:
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- Divide both sides of the equation by $1800 to isolate the compound interest factor on one side.
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- Take the natural logarithm (ln) of both sides to solve for the exponent 12t.
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- Divide by 12 to solve for t.
The calculation will lead us to the number of years, t, rounded to the nearest hundredth of a year.