Question

Let the points \( \left( \frac{11}{2}, \alpha \right) \) lie on or inside the triangle with sides \( x + y = 11 \), \( x + 2y = 16 \), and \( 2x + 3y = 29 \). Then the product of the smallest and the largest values of \( \alpha \) is equal to:

• A. \( 55 \)
• B. \( 33 \)
• C. \( 22 \)
• D. \( 44 \)

Hint:

To solve for the values of \( \alpha \), find the points of intersection of the lines forming the triangle and check the conditions for the points to be inside or on the boundary of the triangle.

Answer 1

The Correct Option is B

A point \( \left( \frac{11}{2}, \alpha \right) \) is located within or on the boundary of a triangle defined by the lines \( x + y = 11 \), \( x + 2y = 16 \), and \( 2x + 3y = 29 \).

Step 1: Define the Triangle Boundaries

The triangle is formed by the intersection of three lines.

• Line 1: \( x + y = 11 \)
• Line 2: \( x + 2y = 16 \)
• Line 3: \( 2x + 3y = 29 \)

Step 2: Determine the Range of \( \alpha \)

Solving the systems of linear equations for these lines will establish the vertices of the triangle. This will allow us to find the minimum and maximum possible values for \( \alpha \), the y-coordinate of the given point \( \left( \frac{11}{2}, \alpha \right) \), such that the point lies within or on the triangle's perimeter.

Step 3: Calculate the Product of Extremal \( \alpha \) Values

After identifying the smallest and largest possible values for \( \alpha \), their product is computed.

The computed product of the smallest and largest values of \( \alpha \) is \( 33 \).

Final Answer: \( 33 \).