The function U(t)=-0.338t2+1.807t+6.545 models the annual unemployment percentage rate for the Phoenix-Mesa-Scottsdale area from the beginning of 2008 through the end of 2014, where t represents the number of years since 2008.

a) Use a graphing method to determine the years in which the unemployment rate was greater that 6.25%

Write your answer in interval notation rounding to one decimal place. 

Complete the following sentence using the information from your graph.

The unemployment rate was above 6.25% starting in year Correct and ending in the year 

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.

The function U(t) = -0.338t^2 + 1.807t + 6.545 models the annual unemployment percentage rate for the Phoenix-Mesa-Scottsdale area from 2008 to 2014. Use a graphing method to determine the years in which the unemployment rate was greater than 6.25%. Write your answer in interval notation rounding to one decimal place. Complete the sentence: 'The unemployment rate was above 6.25% starting in year ____ and ending in the year ____.'

Learn how to solve the quadratic function U(t) = -0.338t^2 + 1.807t + 6.545 to determine when the unemployment rate in the Phoenix-Mesa-Scottsdale area was greater than 6.25% from 2008 to 2014. This problem involves solving quadratic inequalities and applying the quadratic formula, suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation