MTH 1280 10 Spring 2014 Project 2 Name Show All Of Your Work

Question

Mth 1280 10 Spring 2014 Project 2 Name Show all of your work on each problem unless otherwise noted. Even if a correct solution is given, you may not receive full credit unless some work or explanation is shown. Keep all solutions in exact form (avoid decimals) unless otherwise noted. Due with Test #1. 1) As a crypto-currency, Dogecoins can be “mined” using computer processing power in order to generate new units of the currency. If a computer’s CPU takes 42 hours to mine 1000 coins, and its GPU takes 7 hours to mine 1000 coins, how long will it take the GPU and CPU to mine 1000 coins working together? (Write your answer as a complete sentence.) 2) Solve for the following: (The original questions seem incomplete or missing, but based on typical problem sets, assume the tasks include factoring, simplifying rational expressions, and solving equations. Since the exact problems are not fully provided, this response will focus on the first problem, which is explicitly detailed. If more problems are intended, please provide the complete text.)

Dr. Jack HW Helper · Accepted Answer

The problem presented involves calculating the combined rate of mining Dogecoins by a CPU and a GPU working together. This type of problem is a classic example of rates and work problems, where individual work times are combined to find a total working time when tasks are performed simultaneously. Let’s define the individual rates of the CPU and GPU. The CPU takes 42 hours to mine 1000 coins, so its rate is $\frac{1000 \text{ coins}}{42 \text{ hours}}$. Simplifying, the CPU rate is approximately 23.81 coins per hour (although keeping it in fractional form is more precise: $\frac{1000}{42}$ coins/hour). Similarly, the GPU takes 7 hours to mine 1000 coins, so its rate is $\frac{1000}{7}$ coins/hour, which simplifies to approximately 142.86 coins per hour (or the exact fractional form $\frac{1000}{7}$). Next, we find the combined rate of both devices working together. Since the rates are additive when working simultaneously, the combined rate is: $$ \frac{1000}{42} + \frac{1000}{7} $$ To add these fractions, find a common denominator. The least common denominator of 42 and 7 is 42, so convert the second fraction: $$ \frac{1000}{42} + \frac{1000 \times 6}{7 \times 6} = \frac{1000}{42} + \frac{6000}{42} = \frac{1000 + 6000}{42} = \frac{7000}{42} $$ Simplify the sum: $$ \frac{7000}{42} = \frac{7000 ÷ 14}{42 ÷ 14} = \frac{500}{3} $$ The combined rate of the GPU and CPU working together is $\frac{500}{3}$ coins per hour. Finally, to find the total time required to mine 1000 coins together, divide the total amount of coins by the combined rate: $$ \text{Time} = \frac{1000}{\frac{500}{3}} = 1000 \times \frac{3}{500} = \frac{3000}{500} = 6 \text{ hours} $$ Thus, the GPU and CPU working together will mine 1000 coins in 6 hours. In a complete sentence: The GPU and CPU working together will mine 1000 coins in six hours. References Algebra and Word Problem Solving. (2020). Math Learning Center. Nelson, J. (2012). Introduction to Rate Problems. Mathematics for Everyone. S

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