Lec36WS (2)
.pdf
School
Western University**We aren't endorsed by this school
Course
CALCULUS 1000
Subject
Mathematics
Date
Dec 16, 2024
Pages
4
Uploaded by MegaBoulder6508
Math 106Lecture 36 - 12/06/24ReviewWorksheet Instructions:Read through each problem on the worksheet and identify what type of calculustechnique or theorem is needed to solve the problem. Once you have done this for each problem, go back andsolve them!1. Oysters are an important part of the ecosystem in the Chesapeake Bay. They filter out excess mineralslike Nitrogen and provide a habitat to many species including blue crabs. Unfortunately, due to over-fishing and poor water quality their population declined significantly. In 2009, President Barak Obamasigned Executive Order 13508, Chesapeake Bay Protection and Restoration. In 2010, groups startedworking on growing the population of oysters in the bay.Suppose the population over time of Eastern Oysters in Baltimore’s Inner Harbor is given by thefunctionP(t) = 1000 + 2000 sin(t) + 2500twheretis measured in years since 2010. Was there a timewhen the population of Eastern Oysters in the bay was exactly 20,000?2. At an aquarium, scientists are trying to develop jackets for puffer fish with skin injuries. When pufferfish are threatened they puff up into a spherical shape. To make a functional design, the scientists needto know how quickly the jacket fabric will be stretched, which means they need to know how quicklythe surface area is increasing. Assume that the average puffer fish (which can be modeled by a sphere)increases in radius at a speed ofdrdt= 1.5 in/s. How quickly is the surface area growing when the radiusof the fish is 2 in?1
3. You have decided to try your hand at entrepreneurship and open an ice cream shop. How much shouldyou charge for an ice cream cone? In the following questions,X≤$1.50 is the amount we are chargingfor an ice cream cone, andN(X) is the amount of ice cream cones we can sell at that price.(a) Your market research tells you that you can sell 1,000 ice cream cones at $1.50 per cone, and thatyou can sell 1,000 more ice cream cones for each $0.10 reduction in price (for example, you couldsell 3,000 ice cream cones if you charge $1.30 for an ice cream cone). Write an equation forN(X).(b) It takes a fixed up-front cost of $2,000 to set up the ice cream shop. It costs $0.50 to make anice cream cone. Write an equation forC(X) the cost of sellingN(X) ice cream cones at priceX.Substitute the equation you found forN(X) in part (a) into this equation. (Hint: Don’t forgetthe start-up cost!)(c) Write an equation forR(X) your total revenue from sellingN(X) ice cream cones.(Again,substitute in the equation forN(X) found in part (a).)(d) Now that we know the cost and revenue, write an equation forP(X), your profit from sellingN(X) ice cream cones.(e) Finally, determine at what price you should sell your ice cream cones to maximize your profits!2
4. When expanding adiabatically (i.e. without gaining or losing heat), the pressurePand volumeVof asample of argon gas are related by the formulaPV5/3=cwherecis some number dependent on theroom temperature which is constant over time. Suppose the volume of the sample is increasing at arate of .4 liters/sec. How quickly is the pressure decreasing when the volume is 8 liters and pressure is60 kPa?5. When some porcelain dishes are heated up too quickly, the clay body is stressed and can crack. Youare gifted a porcelain tea pot and told that if the temperature changes by more that 30◦F a minute, theteapot will crack. If the temperature of the room is 72◦F and the tea pot starts at room temperatureand reaches 180◦F 3 minutes after water is added, will it crack? How do you know?6. In a depleted frog population, there are two expressions of a gene: spotted and striped. The proportionof frogs that are spotted is represented byp. The average fitness level of the frog population is givenbyw(p) =p2+ 2p(1−p) + (1−p)2. If scientists working to breed frogs to revitalize the frog populationwant to maximize the frog population’s fitness level, what proportion of frogs should they try to makespotted?3
7. It is estimated that the number of new houses that will start being constructed,N(t) (in units of amillion) over the next 5 years is related to the mortgage rater(t) (percent per year) by the equation8N2+r= 36.What is the rate of change of the number of new houses started with respect to timewhen the mortage rate is 4% per year and is increasing at the rate of 0.25% per year?8. Explain how we know that at any time there are two points antipodal (opposite) to each other on theequator which have the same temperature. (Hint: Show there is a place whereT1−T2= 0.)4