Problem 1

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Question: A tech start-up has a budget of $10,000 to buy two types of laptops: Type A (Standard) for $400 each and Type B (Pro) for $700 each. The company must buy at least 15 laptops in total. Additionally, the number of Pro laptops cannot exceed the number of Standard laptops.

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1. Write a system of inequalities to model this situation.
2. Graphically identify the feasible region.
3. If the company wants to maximize the total processing power, where $P = 2A + 5B$, find the number of each laptop they should buy.

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Problem 2

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Question: A drone (Object 1) is flying in a path modeled by the quadratic equation $y = -x^2 + 8x - 5$ . A second drone (Object 2) is flying in a straight line modeled by $y = 2x + 3.$

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1. Find the coordinates $(x, y)$ where the paths of the two drones intersect.
2. Determine the distance between these two intersection points.
3. Use the discriminant of the combined equation to prove whether a third drone flying at $y = 2x + 10$ would ever collide with the first drone's path.

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Problem 3

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Question: A triangular region $R$ is defined by the following system:

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1. Shade the region $R$ on a coordinate plane.
2. Calculate the exact area of the shaded region $R$.
3. Determine if the point $(1, 4)$ is a solution to the system. Justify your answer algebraically.

Problem 4

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Question: A rectangle has a length $L$ and width $W$. The perimeter is measured as $40\text{ cm}$ (rounded to the nearest $10\text{ cm}$). The length is exactly $5\text{ cm}$ longer than the width.

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1. Set up a system of equations to find $L$ and $W$ based on the nominal perimeter.
2. Calculate the Lower Bound for the area of this rectangle, taking into account the rounding of the perimeter.

Answer key

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