The problem involves using an exponential function to calculate the accrued value of an investment. The given equation is $A = 9250(1.063)^t$, where $A$ is the accrued value, and $t$ is the length of the investment in years. We need to find the accrued value after 11 years and the time it takes to reach $40,081.14.

1. Problem Description

The problem involves using an exponential function to calculate the accrued value of an investment. The given equation is $A = 9250(1.063)^t$, where $A$ is the accrued value, and $t$ is the length of the investment in years. We need to find the accrued value after 11 years and the time it takes to reach $40,081.

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2. Solution Steps

Part 1: Find the amount after 11 years.

We substitute $t = 11$ into the equation:

$A = 9250(1.063)^{11}$

$A = 9250 \times (1.063)^{11}$

$A = 9250 \times 2.001437$

$A = 18513.33475$

Round to two decimal places: $A = 18513.33$

Part 2: Find the number of years it takes to reach $40,081.

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4. We set $A = 40081.14$ and solve for $t$:

$40081.14 = 9250(1.063)^t$

Divide both sides by 9250:

$\frac{40081.14}{9250} = (1.063)^t$

$4.3331 = (1.063)^t$

Take the natural logarithm of both sides:

$\ln(4.3331) = \ln((1.063)^t)$

$\ln(4.3331) = t \ln(1.063)$

$t = \frac{\ln(4.3331)}{\ln(1.063)}$

$t = \frac{1.4665}{0.0611}$

$t = 23.999 \approx 24$

3. Final Answer

After 11 years, Jolene will have $18513.33 in her savings account.

Jolene will have accrued $40,081.14 in her savings after 24 years.