We are given that beef consumption is modeled by exponential decay, meaning the form of the equation will be:

$B(t) = B_0 e^{-kt}$ where:

• $B(t)$ is the beef consumption after $t$ years since 2000,
• $B_0$ is the initial consumption (in the year 2000),
• $k$ is the decay constant,
• $t$ is the time in years since 2000.

Part (a): Find the value of $k$ and write the equation

We are given:

• $B(0) = 65.8$ lb in the year 2000,
• $B(6) = 60.5$ lb in the year 2006 (which is 6 years after 2000).

To find $k$, we use the fact that $B(6) = 60.5$. Plug this into the exponential decay formula:

$B(6) = B_0 e^{-6k} = 65.8 e^{-6k}$

Substitute $B(6) = 60.5$:

$60.5 = 65.8 e^{-6k}$

Now, solve for $k$:

1. Divide both sides by 65.8:

$\frac{60.5}{65.8} = e^{-6k}$

2. Take the natural logarithm (ln) of both sides:

$\ln\left(\frac{60.5}{65.8}\right) = -6k$

3. Solve for $k$:

$k = -\frac{1}{6} \ln\left(\frac{60.5}{65.8}\right)$

Now, calculate $k$.

Let me compute that for you.The value of $k$ is approximately $0.0140$ (rounded to four decimal places).

Thus, the equation for the beef consumption is:

$B(t) = 65.8 e^{-0.0140t}$

Part (b): Estimate the consumption of beef in 2011

To estimate the beef consumption in 2011, which is $t = 11$ years after 2000, we substitute $t = 11$ into the equation:

$B(11) = 65.8 e^{-0.0140 \times 11}$

Let me calculate that for you.The estimated beef consumption in 2011 is approximately 56.41 pounds per person.

Part (c): In what year will the consumption of beef be 20 pounds?

We need to solve for $t$ when $B(t) = 20$. Using the equation $B(t) = 65.8 e^{-0.0140t}$, we set $B(t) = 20$:

$20 = 65.8 e^{-0.0140t}$

Solving for $t$:

1. Divide both sides by 65.8:

$\frac{20}{65.8} = e^{-0.0140t}$

2. Take the natural logarithm (ln) of both sides:

$\ln\left(\frac{20}{65.8}\right) = -0.0140t$

3. Solve for $t$:

$t = -\frac{1}{0.0140} \ln\left(\frac{20}{65.8}\right)$

Let me compute that for you.The beef consumption is projected to reach 20 pounds per person approximately 85 years after 2000. This corresponds to the year:

$2000 + 85 = 2085$

Thus, the consumption of beef is estimated to be 20 pounds in the year 2085.

Summary:

• a) The value of $k$ is approximately $0.0140$, and the equation is $B(t) = 65.8 e^{-0.0140t}$.
• b) The estimated beef consumption in 2011 is about 56.41 pounds per person.
• c) The consumption is projected to reach 20 pounds in the year 2085.

Would you like more details on any part of this solution? Here are some related questions for practice:

1. What would the beef consumption be in 2025 according to this model?
2. How does the consumption change between 2010 and 2020?
3. What would happen if the decay rate were faster, say $k = 0.02$?
4. How long would it take for consumption to drop to 10 pounds per person?
5. What is the half-life of beef consumption under this model?

Tip: Exponential decay models are often used in various fields like population studies, radioactive decay, and economics to model diminishing quantities.