Final answer:
To find out how many years it will take for Holly's investment to quadruple, we use the compound interest formula and solve for the number of years (t). By substituting the given values and solving the equation, we determine it will take approximately 27.3 years for her investment to quadruple in value.
Step-by-step explanation:
The question asks how many years it will take for an investment of $1,600 at a 5.2% interest rate compounded weekly to quadruple in value. To solve this, we can use the compound interest formula A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years.
We know that A should be 4 times the initial investment (4*$1,600 = $6,400), P is $1,600, r is 0.052 (5.2%), and n is 52 (since the interest is compounded weekly). We are looking for t when A is $6,400.
First, we need to set up the equation using these values:
- $6,400 = $1,600(1 + 0.052/52)^(52t)
Next, divide both sides by $1,600:
Now, take the natural logarithm of both sides:
- ln(4) = ln((1 + 0.052/52)^(52t))
Use the power rule of logarithms to bring down the exponent:
- ln(4) = 52t * ln(1 + 0.052/52)
Divide both sides by 52 * ln(1 + 0.052/52) to solve for t:
- t = ln(4) / (52 * ln(1 + 0.052/52))
Use a calculator to find t:
It will take approximately 27.3 years for Holly's investment to be worth quadruple her initial investment.