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The question here is about applying a quadratic function to analyze beef consumption over time.

Given:
\[
U(t) = 5t^2 - 20t + 20759
\]
where \( t \) is the number of years since 2000 (so \( t = 0 \) represents the year 2000, \( t = 1 \) represents 2001, etc.), and \( U(t) \) represents the annual beef consumption in tons.

The problem has two parts:

### Part (a)
You need to determine when the annual consumption of beef was **above 21300 tons per year**. To do this:
1. Set up the inequality \( U(t) > 21300 \).
2. Substitute \( U(t) = 5t^2 - 20t + 20759 \) and solve:
   \[
   5t^2 - 20t + 20759 > 21300
   \]
3. Simplify this to find the range of \( t \) values satisfying the inequality.

### Part (b)
This part asks when the annual consumption of beef was **below 21380 tons per year**:
1. Set up the inequality \( U(t) < 21380 \).
2. Substitute \( U(t) = 5t^2 - 20t + 20759 \) and solve:
   \[
   5t^2 - 20t + 20759 < 21380
   \]
3. Simplify this to find the range of \( t \) values that satisfy this inequality.

For both parts, you will graph the function \( U(t) \) and look for the points where the function crosses the values 21300 and 21380. This intersection will give the time intervals you're looking for. The answers should be expressed in interval notation, rounding to one decimal place as requested.

Would you like the detailed steps on solving these inequalities? 

Here are five related questions to expand understanding:
1. How do you solve quadratic inequalities algebraically without graphing?
2. What is interval notation, and why is it useful in expressing ranges?
3. How do you interpret the significance of the coefficients in a quadratic function in a real-world context?
4. What methods can be used to estimate the intersection points if exact algebraic solutions are difficult?
5. How does adjusting the viewing window on a calculator help with graphing complex functions?

**Tip:** When using a graphing calculator, always adjust the viewing window based on the problem's given range to capture the relevant section of the graph accurately.

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 21300 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 21380 tons per year. Write your answer in interval notation rounding to one decimal place.

Explore how to use a quadratic function to analyze the years when beef consumption in the U.S. was above 21300 tons and below 21380 tons. This question uses quadratic inequalities, graphing techniques, and interval notation to provide solutions, suitable for students in Grades 9-12.

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