Final answer:
To find the number of years it will take for $7000 to grow to $12000 when invested at a continuous compound interest rate of 3.50%, we can use the formula: A = P * e^(rt). By substituting the values into the formula and solving for t, we find that it will take approximately 9.7557 years for $7000 to grow to $12000.
Step-by-step explanation:
To find the number of years it will take for $7000 to grow to $12000 when invested at a continuous compound interest rate of 3.50%, we can use the formula:
A = P * e^(rt)
Where:
- A is the final amount (which is $12000)
- P is the initial principal (which is $7000)
- r is the continuous interest rate (which is 3.50% or 0.035 in decimal form)
- t is the number of years
By substituting the values into the formula, we get:
$12000 = $7000 * e^(0.035t)
To solve for t, we need to isolate it. Divide both sides of the equation by $7000:
e^(0.035t) = $12000 / $7000
Simplify the right side:
e^(0.035t) = 1.7143
To isolate t, we need to take the natural logarithm of both sides:
ln(e^(0.035t)) = ln(1.7143)
Using the property of logarithms, the exponent comes down as a coefficient:
0.035t * ln(e) = ln(1.7143)
ln(e) is equal to 1, so the equation becomes:
0.035t = ln(1.7143)
To solve for t, divide both sides by 0.035:
t = ln(1.7143) / 0.035
Using a calculator or computer program, we find that t is approximately 9.7557 years.