The expression 1.0.0341 models the balance, in thousands of dollars, where t represents time in years after the account was opened.

a. What does the 0.034 represent in this context?

b. Write an expression for the number of years after which there will be 15,000 dollars in the account.

Answer

To address the query regarding the expression 1.0.0341, which models the balance in thousands of dollars with respect to time in years after an account is opened, we can analyze the components of the expression step by step.

a. What does the 0.034 represent in this context?

The expression can be interpreted in terms of a mathematical model, typically of the form:

[ B(t) = B_0 + rt ]

where:

In the expression "1.0.0341", we can break it down:

Thus, 0.034 in this context indicates that the account grows by approximately $34 (or 0.034 in thousands) for each year that the account is open.

b. Write an expression for the number of years after which there will be 15,000 dollars in the account.

To find the number of years until the balance reaches $15,000, we can set up the equation based on our expression.

We know:

Now, we want to find ( t ) when ( B(t) = 15 ):

[ 15 = 1 + 0.034t ]

Subtracting 1 from both sides:

[ 14 = 0.034t ]

Now, we can solve for ( t ) by dividing both sides by 0.034:

[ t = \frac{14}{0.034} \approx 411.76 ]

Therefore, the expression for the number of years after which there will be $15,000 in the account is approximately 411.76 years.

This tells us that under the given model, it would take a little over 411 years for the balance to reach $15,000 at the specified growth rate.