The problem is to solve for $t$ in the equation $6.63 = 15[1 - e^{-t/10}]$.

1. Problem Description

The problem is to solve for $t$ in the equation $6.63 = 15[1 - e^{-t/10}]$.

2. Solution Steps

We are given the equation:

$6.63 = 15[1 - e^{-t/10}]$

First, divide both sides of the equation by 15:

$\frac{6.63}{15} = 1 - e^{-t/10}$

$0.442 = 1 - e^{-t/10}$

Next, isolate the exponential term by subtracting 1 from both sides:

$0.442 - 1 = -e^{-t/10}$

$-0.558 = -e^{-t/10}$

Multiply both sides by -1:

$0.558 = e^{-t/10}$

Now, take the natural logarithm (ln) of both sides:

$ln(0.558) = ln(e^{-t/10})$

$ln(0.558) = -\frac{t}{10}$

Multiply both sides by -10:

$-10 \cdot ln(0.558) = t$

$t = -10 \cdot ln(0.558)$

$t \approx -10 \cdot (-0.5836)$

$t \approx 5.836$

3. Final Answer

The value of $t$ is approximately $5.836$.