To solve this problem, we need to find the interval where the function $U(t) = -0.338t^2 + 1.807t + 6.545$ is greater than 6.25%. This will involve finding the roots of the equation $U(t) = 6.25$ and determining when the function is above this value.

Step 1: Set up the equation

We want to solve: $-0.338t^2 + 1.807t + 6.545 = 6.25$ This simplifies to: $-0.338t^2 + 1.807t + 0.295 = 0$

Step 2: Solve the quadratic equation

The quadratic equation can be written as: $at^2 + bt + c = 0$ where $a = -0.338$, $b = 1.807$, and $c = 0.295$.

The quadratic formula is given by: $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substitute the values of $a$, $b$, and $c$: $t = \frac{-1.807 \pm \sqrt{1.807^2 - 4(-0.338)(0.295)}}{2(-0.338)}$

Calculating each part:

1. $b^2 = 1.807^2 = 3.265$
2. $4ac = 4(-0.338)(0.295) = -0.3986$
3. $b^2 - 4ac = 3.265 + 0.3986 = 3.6636$

Now, calculate the square root: $\sqrt{3.6636} \approx 1.914$

Plug these values back into the quadratic formula: $t = \frac{-1.807 \pm 1.914}{2(-0.338)}$

Calculating the two roots:

1. $t_1 = \frac{-1.807 + 1.914}{-0.676} \approx -0.158$
2. $t_2 = \frac{-1.807 - 1.914}{-0.676} \approx 5.5$

Step 3: Interpret the results

The roots are approximately $t \approx -0.2$ and $t \approx 5.5$, which correspond to the years where the unemployment rate crosses 6.25%. Since $t = -0.2$ is before 2008, we only consider $t = 5.5$, indicating that the unemployment rate was above 6.25% from the start of 2008 until halfway through 2013.

Final Answer

The unemployment rate was above 6.25% starting in year $0$ (2008) and ending in year $5.5$ (mid-2013). In interval notation, this is: $[0, 5.5]$

Would you like more details or have any questions? Here are some related questions to consider:

1. How do we interpret the roots of a quadratic equation in real-world scenarios?
2. What methods can be used to solve quadratic inequalities graphically?
3. How would the function change if the coefficients were altered slightly?
4. What does the vertex of the quadratic function represent in this context?
5. How can we use derivatives to find critical points of the quadratic function?

Tip: When solving quadratic equations, always check if the discriminant ($b^2 - 4ac$) is positive, zero, or negative, as it determines the nature of the roots.