We are given the equation:

\[
6 = 0.167t^3 - 2t^2 + 4.832t + 7
\]

We need to solve for \( t \). To simplify the equation, we first move the 6 to the other side:

\[
0 = 0.167t^3 - 2t^2 + 4.832t + 7 - 6
\]

\[
0 = 0.167t^3 - 2t^2 + 4.832t + 1
\]

Now, we have a cubic equation:

\[
0.167t^3 - 2t^2 + 4.832t + 1 = 0
\]

Since this is a cubic equation, we can solve it using numerical methods or approximation techniques such as the Newton-Raphson method, or graphically determine the roots. I'll compute the solutions numerically.

Let me calculate it.The solutions to the cubic equation \( 0.167t^3 - 2t^2 + 4.832t + 1 = 0 \) are approximately:

\[
t_1 \approx -0.192
\]
\[
t_2 \approx 3.686
\]
\[
t_3 \approx 8.481
\]

These are the real solutions to the equation.

Would you like more details on how these solutions were found or have any further questions? Here are some related questions for exploration:

1. How are cubic equations solved algebraically versus numerically?
2. What is the significance of having three real roots in a cubic equation?
3. How can we verify solutions to a cubic equation?
4. What are common methods for solving polynomial equations?
5. How does the Newton-Raphson method work for solving equations?

**Tip**: When dealing with cubic equations, graphical methods often give good insights into the behavior of the function before solving it numerically.

Explore how to solve the cubic equation 0.167t^3 - 2t^2 + 4.832t + 1 = 0 using numerical methods. This solution covers important algebraic concepts like cubic equations, root-finding techniques, and the Fundamental Theorem of Algebra. Suitable for students in Grades 11-12.

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