Mathematics is the cornerstone of logic and reasoning, shaping the way we solve problems in various fields such as engineering, finance, and data science. At www.mathsassignmenthelp.com, we aim to bridge the gap between complex theories and practical applications by offering top-notch Math Assignment Help. Today, we present two master-level mathematics questions solved by our experts to showcase the depth of our assistance.

Question 1: Optimization in Real-Life Applications

Question: A company produces two types of products, A and B. The profit from each product is $50 for A and $70 for B. Manufacturing constraints require that no more than 200 units of A and 300 units of B are produced weekly. Additionally, the combined total of products manufactured should not exceed 400 units due to limited resources. How should the company allocate production to maximize profit?

Solution:

This question is a classic linear programming problem that requires optimization under constraints. Let’s break it down step by step:

1. Define the variables: Let represent the number of units of Product A and represent the number of units of Product B.

2. Objective function: The goal is to maximize profit, given by:

3. Constraints: The problem specifies the following:

   • (Product A limit)

   • (Product B limit)

   • (Total production limit)

   • (Non-negative production)

4. Graphical solution: Plot the constraints on a graph. The feasible region, where all conditions are satisfied, is a polygon formed by the intersection of these constraints.

5. Evaluate corner points: The vertices of the feasible region are determined by solving the equations of the constraints:

   • (0, 0): Profit = 0

   • (0, 300): Profit = 70(300) = 21,000

   • (200, 200): Profit = 50(200) + 70(200) = 24,000

   • (200, 0): Profit = 50(200) = 10,000

6. Optimal solution: The maximum profit occurs at (200, 200), meaning the company should produce 200 units of A and 200 units of B for a weekly profit of $24,000.

Our Math Assignment Help services can guide you through such optimization challenges, providing you with practical strategies and accurate results.

Question 2: Probability and Decision-Making

Question: A quality control inspector checks a batch of 100 light bulbs. Each bulb has a 95% probability of functioning correctly. If more than 90 bulbs function correctly, the batch is approved. What is the probability that the batch will be approved?

Solution:

This problem delves into probability distributions, specifically the binomial distribution, to evaluate the likelihood of approval.

1. Define the problem:

   • Number of trials

   • Success probability

   • Approval condition: More than 90 bulbs function correctly ()

2. Use the binomial distribution formula:

   However, calculating directly for a large is computationally intense.

3. Apply the normal approximation: For large , the binomial distribution approximates the normal distribution with:

   • Mean

   • Standard deviation

4. Standardize the condition: Convert to the standard normal variable : (0.5 is added for continuity correction).

5. Find the probability: Using standard normal distribution tables, .

6. Conclusion: The probability that the batch will be approved is approximately 98%.

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