To transform the function $P(t) = 0.5t^2 - 9.65t + 100$ into a different form with $P(t) = 200$, we can set the equation equal to 200:

$0.5t^2 - 9.65t + 100 = 200$

Subtract 200 from both sides:

$0.5t^2 - 9.65t + 100 - 200 = 0$

This simplifies to:

$0.5t^2 - 9.65t - 100 = 0$

Now, to solve for $t$, we can multiply the entire equation by 2 to eliminate the decimal:

$t^2 - 19.3t - 200 = 0$

Next, we can apply the quadratic formula:

$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = 1$, $b = -19.3$, and $c = -200$:

1. Calculate the discriminant:

$b^2 - 4ac = (-19.3)^2 - 4(1)(-200) = 372.49 + 800 = 1172.49$

2. Substitute into the quadratic formula:

$t = \frac{19.3 \pm \sqrt{1172.49}}{2}$

3. Calculate the square root and the two potential values for $t$.

Would you like to see the calculations for $t$ or have any specific questions about this transformation? Here are some related questions you might consider:

1. What are the exact solutions for $t$?
2. How can we interpret the physical meaning of $P(t)$ in this context?
3. Can we graph this quadratic function to visualize the changes?
4. What happens to $P(t)$ if the coefficients change?
5. How do you find the vertex of the quadratic function?

Tip: Always check your discriminant to determine the nature of the roots in a quadratic equation!