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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 AlgebraA company produces two products, A and B. Each product requires labor from twodepartments: machining and assembly.●Product A requires 3 hours of machining and 2 hours of assembly.●Product B requires 4 hours of machining and 5 hours of assembly.●The machining department has 240 available hours, and the assembly departmenthas 200 available hours.How many of each product should the company produce to fully utilize its availablehours?2. CalculusA rectangular sheet of metal is 10 cm by 20 cm. Squares of side xxx cm are cut from eachcorner and the sides are folded up to form an open box.●Write an equation for the volume of the box as a function of xxx.●Determine the value of xxx that maximizes the volume of the box.3. ProbabilityA bag contains 10 red balls, 8 blue balls, and 6 green balls. Three balls are drawn atrandom without replacement.●What is the probability that all three balls are red?●What is the probability that one ball of each color is drawn?4. Differential EquationsThe population of a certain species grows according to the logistic equation:dPdt=0.03P(1−P500),\frac{dP}{dt} = 0.03P\left(1 - \frac{P}{500}\right),dtdP=0.03P(1−500P),where P(t)P(t)P(t) is the population at time ttt (in years).●Find the population P(t)P(t)P(t) when P(0)=100P(0) = 100P(0)=100.
●Determine the population as t→∞t \to \inftyt→∞.5. OptimizationA farmer has 1,000 meters of fencing to enclose a rectangular field.●What dimensions will maximize the area of the field?●If one side of the field is along a river and does not require fencing, what are thenew dimensions to maximize the area?6. FinanceYou deposit $10,000 in an account that earns 5%5\%5% annual interest, compoundedquarterly.●Write a formula to calculate the amount of money in the account after ttt years.●Determine how much money will be in the account after 10 years.7. GeometryA cone has a height of 12 cm and a base radius of rrr cm.●Write a formula for the surface area of the cone in terms of rrr.●If the surface area is 300 cm², find the value of rrr.8. Systems of EquationsA student spends $12 on a combination of pencils and notebooks. Each pencil costs$0.50, and each notebook costs $2.●Write a system of equations to represent the situation.●Solve the system to find how many pencils and notebooks the student bought.