The function U(t)=5t2-20t+20759 models the annual consumption of beef, in tons, in the U.S from 2000 to 2017. where t represents the number of years since 2000.

Note: For the problems below, set the window on your calculator to x: [0..18] and y: [20000..22000]

a) Use a graphing method to determine the years in which the consumtion of beef was greater that 20980 tons per year.

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

Complete the following sentence using the information from your graph.

The annual consumption of beef was above 20980 tons per year starting in year .

b) Use a graphing method to determine the years in which the annual consumption of beef was less than 20975 tons per year.

Write your answer in interval notation rounding to one decimal place

To answer the given questions, we will follow these steps:

### The function given is:  
\[ U(t) = 5t^2 - 20t + 20759 \]
where \( t \) represents the number of years since 2000, and \( U(t) \) is the annual consumption of beef in tons.

We are tasked with using a graphing method to find specific intervals where the consumption was above or below certain levels. We can visualize this function using a calculator or graphing tool and follow the instructions for the window settings:  
- \( t \)-axis (x-axis): \([0, 18]\)  
- \( U(t) \)-axis (y-axis): \([20000, 22000]\)

### Part (a): Consumption of beef greater than 20980 tons

We are looking for the values of \( t \) where \( U(t) > 20980 \).

1. Graph \( U(t) = 5t^2 - 20t + 20759 \) and the horizontal line \( U(t) = 20980 \).
2. Find the points of intersection between the parabola and the horizontal line. These are the values of \( t \) where \( U(t) = 20980 \).
3. From the graph, observe the intervals where the parabola is above \( U(t) = 20980 \).

By solving this graphically or algebraically, we can approximate the values of \( t \) that correspond to the intersections. After finding those points, the interval where \( U(t) > 20980 \) will be:

- \( t \in [t_1, t_2] \), where \( t_1 \) and \( t_2 \) are the years when beef consumption was exactly 20980 tons.

The **annual consumption of beef was above 20980 tons per year starting in year** \( t_1 \).

### Part (b): Consumption of beef less than 20975 tons

Here we are looking for the values of \( t \) where \( U(t) < 20975 \).

1. Graph \( U(t) = 5t^2 - 20t + 20759 \) and the horizontal line \( U(t) = 20975 \).
2. Find the points of intersection between the parabola and the horizontal line. These are the values of \( t \) where \( U(t) = 20975 \).
3. Observe the intervals where the parabola is below \( U(t) = 20975 \).

The interval where \( U(t) < 20975 \) will be in a specific range of \( t \). By graphing or solving for the intersections, you'll find the interval in which the annual consumption was below 20975 tons.

### Answer format:
- Part (a) interval in the form \([t_1, t_2]\)
- Part (b) interval in the form \([t_1, t_2]\)

Would you like help with the graphing process or further details on solving the quadratic equation? Here are 5 related questions for more practice:

1. What are the years when beef consumption was exactly 20980 tons per year?
2. How can you solve the inequality \( 5t^2 - 20t + 20759 > 20980 \) algebraically?
3. What does the vertex of the parabola \( U(t) = 5t^2 - 20t + 20759 \) represent in this context?
4. How would changing the window settings affect the clarity of the graph?
5. What is the minimum annual beef consumption during the years 2000 to 2017?

**Tip**: When graphing quadratic functions, the vertex helps you understand the turning point and whether the function is increasing or decreasing in specific intervals.

The function U(t)=5t^2-20t+20759 models the annual consumption of beef, in tons, in the U.S from 2000 to 2017. Where t represents the number of years since 2000. a) Use a graphing method to determine the years in which the consumption of beef was greater than 20980 tons per year. Write your answer in interval notation rounding to one decimal place. b) Use a graphing method to determine the years in which the annual consumption of beef was less than 20975 tons per year. Write your answer in interval notation rounding to one decimal place.

Explore how to solve quadratic inequalities using graphing methods with the function U(t) = 5t^2 - 20t + 20759, which models beef consumption in the U.S. from 2000 to 2017. Learn to find the intervals where consumption exceeds 20980 tons and falls below 20975 tons. Suitable for Grades 9-12.

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