(a) Find $E'(t)$

The given function for fuel efficiency is:

$E(t) = -0.0007t^3 + 0.0278t^2 - 0.0843t + 12$

To find $E'(t)$, we take the derivative of each term:

1. The derivative of $-0.0007t^3$ is $-0.0007 \cdot 3t^2 = -0.0021t^2$
2. The derivative of $0.0278t^2$ is $0.0278 \cdot 2t = 0.0556t$
3. The derivative of $-0.0843t$ is $-0.0843$
4. The derivative of the constant $12$ is $0$

Thus, the derivative $E'(t)$ is:

$E'(t) = -0.0021t^2 + 0.0556t - 0.0843$

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(b) Find the values of $a$ and $b$

Fuel efficiency is increasing when $E'(t) > 0$. To find when this occurs, we need to solve $E'(t) > 0$.

The equation is:

$-0.0021t^2 + 0.0556t - 0.0843 > 0$

Let's solve this quadratic inequality.

First, solve the corresponding quadratic equation $-0.0021t^2 + 0.0556t - 0.0843 = 0$ to find the critical points.The critical points are approximately $t = 1.61$ and $t = 24.86$.

Now, to determine when fuel efficiency is increasing, we check the sign of $E'(t)$ in the intervals determined by these critical points:

1. For $t < 1.61$
2. For $1.61 < t < 24.86$
3. For $t > 24.86$

We'll test the sign of $E'(t)$ in these intervals.The results for $E'(t)$ in the test intervals are:

• $E'(t) < 0$ for $t < 1.61$, indicating fuel efficiency is decreasing before $t = 1.61$.
• $E'(t) > 0$ for $1.61 < t < 24.86$, indicating fuel efficiency is increasing between these values of $t$.
• $E'(t) < 0$ for $t > 24.86$, indicating fuel efficiency is decreasing again after $t = 24.86$.

Thus, fuel efficiency was improving between the years:

• $a = 1.61$ (which corresponds to 1971.61, or approximately 1972)
• $b = 24.86$ (which corresponds to 1994.86, or approximately 1995).

Final Answer:

• Fuel efficiency was improving between 1972 and 1995.

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Would you like further details or explanations on any part of this? Here are 5 related questions you might find interesting:

1. How can we determine the maximum fuel efficiency within the interval?
2. What is the significance of $E(t) = 0$ in this context?
3. How can you solve more complex inequalities involving cubic or higher-degree functions?
4. What other real-world applications might use a similar mathematical model?
5. How does calculus help in optimizing real-life systems?

Tip: Quadratic inequalities can often be tested by checking the sign of the expression at various points in its intervals!