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School
Brock University**We aren't endorsed by this school
Course
MATH 1P01
Subject
Mechanical Engineering
Date
Dec 19, 2024
Pages
2
Uploaded by CountFreedom9250
Instructions:Solve each problem. Show all work for full credit.1. Linear ProgrammingA factory produces two types of chairs: standard and deluxe. Each standard chair requires 2hours of cutting and 4 hours of assembly. Each deluxe chair requires 3 hours of cutting and 6hours of assembly.●The factory has 120 hours available for cutting and 240 hours available for assemblyeach week.●The profit on a standard chair is $30, and the profit on a deluxe chair is $50.How many of each type of chair should the factory produce to maximize profit? Formulate andsolve the problem using linear programming.2. CalculusA company’s revenue is modeled by R(x)=100x−0.5x2R(x) = 100x - 0.5x^2R(x)=100x−0.5x2,where xxx is the number of units sold.●Find the number of units that maximizes the company’s revenue.●What is the maximum revenue?3. StatisticsA professor finds that the time students take to complete an exam is normally distributed with amean of 90 minutes and a standard deviation of 15 minutes.●What percentage of students complete the exam in less than 75 minutes?●If the exam lasts 120 minutes, what percentage of students are unable to finish withinthe allotted time?4. ProbabilityA game involves rolling two six-sided dice.●What is the probability of rolling a sum of 8?●If the game is played 100 times, how many times would you expect to roll a sum of 8?
5. OptimizationA cylindrical can is to be made to hold 1 liter of liquid (1,000 cm³).●Write an equation for the surface area of the can in terms of its radius rrr and height hhh.●Find the dimensions of the can (radius and height) that minimize the surface area.6. FinanceYou take out a loan of $20,000 at an annual interest rate of 6%6\%6%, compounded monthly, tobe repaid over 5 years.●Write the formula for the monthly payment.●Calculate the monthly payment.7. GeometryA rectangular piece of cardboard measures 15 cm by 20 cm. Squares of side xxx cm are cutfrom each corner, and the remaining sides are folded to form an open box.●Write a formula for the volume of the box as a function of xxx.●Find the value of xxx that maximizes the volume of the box.8. Differential EquationsA tank contains 50 liters of water and 10 kg of salt. A solution containing 0.5 kg of salt per liter ispumped into the tank at a rate of 5 liters per minute, and the mixture is pumped out at the samerate.●Write the differential equation describing the amount of salt in the tank.●Solve the differential equation to find the amount of salt in the tank at any time ttt.