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SOLVE THE THREE OF THEM. Solve only the questions marked with blue circles.

Let's analyze and solve the problems marked with blue circles in the provided image.

### Problem 107:
**Question:** The number of eggs (in billions) produced in the United States from 1990 to 2002 can be modeled by the quadratic equation \(E(t) = 4.6t^2 + 9.7t + 62.1\), where \(t\) represents the years since 1990.

**Tasks:**
- (a) **Interpret the coefficients:** The coefficients \(4.6\), \(9.7\), and \(62.1\) correspond to the quadratic, linear, and constant terms, respectively. Explain their meanings.
- (b) **Determine the year when egg production was at its minimum:** This corresponds to the vertex of the parabola.
- (c) **Discuss the feasibility of the model if the trend continues indefinitely.**

**Solution:**
- **(a) Interpretation of Coefficients:**
  - \(4.6\): This is the coefficient of the quadratic term \(t^2\), which indicates that egg production is accelerating over time.
  - \(9.7\): This is the coefficient of the linear term \(t\), representing the initial rate of increase in egg production.
  - \(62.1\): This is the constant term, representing the number of eggs produced (in billions) in the base year, 1990.

- **(b) Year of Minimum Production:**
  The vertex of the parabola given by \(E(t) = at^2 + bt + c\) is found at \(t = -\frac{b}{2a}\).
  \[
  t = -\frac{9.7}{2 \times 4.6} \approx -1.054 \text{ years after 1990}
  \]
  Since time \(t\) cannot be negative, the model suggests that the minimum might theoretically occur slightly before 1990. This indicates that 1990 was close to the point of minimum production.

- **(c) Feasibility of Model Indefinitely:**
  If the trend continues, the model predicts a parabolic increase in egg production, which is unrealistic in the long term due to resource constraints, market saturation, and other factors.

### Problem 109:
**Question:** Blood pressure is measured in millimeters of mercury. Determine the interval on the inequality \(P(t) = 1.5t + 80 < 150\), which represents when blood pressure is considered within the normal range.

**Solution:**
Given \(P(t) = 1.5t + 80 < 150\), we can solve for \(t\):

\[
1.5t + 80 < 150
\]
\[
1.5t < 150 - 80
\]
\[
1.5t < 70
\]
\[
t < \frac{70}{1.5} \approx 46.67
\]
Thus, the normal blood pressure range corresponds to the interval \( t < 46.67 \).

### Problem 111:
**Question:** A conductor of length \(L = 100\) cm carrying a current \(I\) generates a magnetic field. If \(I = 10\) A and \(B = 5 \times 10^{-5}\) T, determine the angle \(\theta\) between the current and the magnetic field.

**Solution:**
We can use the formula for the force on a conductor in a magnetic field: \(F = ILB \sin \theta\).

Given that the force \(F\) is not provided, the problem might be to find the angle \(\theta\) under the condition that the entire length of the conductor is perpendicular to the magnetic field (\(\theta = 90^\circ\)). If that's the case, then:
\[
\sin \theta = 1 \quad \Rightarrow \quad \theta = 90^\circ
\]

However, if the angle is not perpendicular, additional information would be needed to find \(\theta\).

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Do you need further details or explanations? Here are five related questions to deepen your understanding:

1. How can you derive the formula for the vertex of a parabola?
2. What real-world factors could alter the egg production model?
3. How would you modify the blood pressure inequality if the threshold changed to 140 mmHg?
4. What happens to the angle \(\theta\) in Problem 111 if the force is doubled?
5. How does changing the length \(L\) of the conductor affect the magnetic force?

**Tip:** Always cross-check the units in physical equations to ensure they align correctly when solving for variables.

Learn how to solve quadratic equations, inequalities, and magnetic fields problems with detailed step-by-step solutions. Covering topics such as interpreting quadratic coefficients, determining normal ranges in inequalities, and calculating angles in magnetic fields. Suitable for high school students.

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