Math 122 Fall 2008 Unit Test 2 Review Problems – Set A We have chosen these problems because we think that they are representative of many of the mathematical concepts that we have studied. There is no guarantee that the problems that appear on the exam will resemble these problems in any way whatsoever. Remember that on exams you will have to supply evidence for your conclusions and may have to explain why your answers are reasonable and appropriate. 1. Corks for wine bottles are usually from strips of bark from the cork tree (Phellodendron amurense). A cork from a typical bottle is a roughly cylindrical shape whose volume can be determined with the formula: ! V ="# r2# h where r is the radius of the cork and h is the height of the cork. Simple cylindrical corks are most suitable for “still” wines. That is, wines that do not have any carbonation due to dissolved CO2 present in the wine. Corks for sparkling wines (the best known being the wines from the Champagne region of France) are designed quite differently (see photograph1). This is because the contents of a bottle of sparkling wine are under very high pressure (often five times regular atmospheric pressure or higher). The cork for a bottle of sparkling wine has to both seal the bottle (the function of any cork) and to resist the strong forces exerted on it by the pressurized contents of the wine bottle. In this problem you will calculate the volume of a champagne cork and the volume of a conventional cork (for the sake of comparison). (a) A conventional cork from a bottle of still wine typically measures about 4.4cm in length with a radius of about 0.9 cm. Calculate the volume of a conventional cork in cubic centimeters. (b) Figure 7 shows a shaded area, which when revolved around the x-axis creates a reasonable approximation for the shape of a cork from a bottle of Perrier-Jouet Brut Imperial Champagne, vintage 1994. Sketch a three dimensional picture of the shape that will be formed by revolving the shaded area from Figure 7 about the x-axis and indicate how you can “slice” this shape up into simpler pieces. (c) The portion of the cork that is nearest to the y-axis can be approximated by a cylinder. Calculate the volume of this portion of the champagne cork in units of cubic centimeters. (d) The outline of the remaining portion of the champagne cork is described by the equation: ! y =512" x +212 valid from x = 2 to x = 5. Set up an integral whose numerical value will equal the volume of this portion of the champagne cork in units of cubic centimeters. 1 Image source: http://www.cyberbacchus.com/ Figure 6: Cork for a bottle of Champagne.0 1 2 3 4 5 0 1 2 Figure 7. (e) Evaluate the integral from Part (d) of this problem. (f) What is the total volume of the champagne cork in units of cubic centimeters? How does this compare to the volume of a conventional cork? 2. A rectangular lake is 150 km long and 3 km wide. The vertical cross-section through the lake is shown in the diagram given below. 0.2 km 3 km (a) Set up an integral for the volume of the lake slicing vertically. (b) Set up an integral for the volume of the lake slicing horizontally. (c) Evaluate the two integrals to find the volume of the lake in cubic kilometers. (You should get the same value each time.) 3. In this problem your objective in each part is to find the volume of revolution when the region described below is rotated around the x-axis. The axis of rotation in each case is the x-axis. (a) The region bounded by: ! y = x2, ! y = 0, ! x = 0, and ! x = 1. (b) The region bounded by: ! y = 4 " x2, ! y = 0, ! x = "2, and ! x = 0. (c) The region bounded by: ! y = ex, ! y = 0, ! x = "1, and ! x = 1. (d) The region bounded by: ! y =1x +1, ! y = 0, ! x = 0, and ! x = 1.(e) The region bounded by: ! y = x2, ! y = x, ! x = 0, and ! x = 1. 4. A triangular object is shown below. The density of the object is ! "x( )= 1+ x g/cm2. x y 1 -1 1 (a) Find the total mass of the object. (b) Do you think that the x-coordinate of the center of mass will be to the left or to the right of the x-axis? Briefly (in a sentence or two) explain why you think your answer is correct. (c) Find the exact value of the x-coordinate of the center of mass for the triangular object. 5. A gas station stores its gasoline in a tank under the ground. The tank is a cylinder lying horizontally on its side. The radius of the cylinder is 4 feet, its length is 12 feet and its top is 10 feet under the ground. Let y = 0 represent the height at the center of the cylindrical portion of the gasoline tank. The gasoline stored inside the tank has flakes of rust in it. As a result, the weight of the gasoline varies throughout the tank. The weight of gasoline (in pounds per cubic foot or lb/ft3) is given by the formula: ! w y( )= 42 " 0.1y when y is measured in feet. If the tank starts out completely full of gasoline, find the total work needed to pump all of the gasoline out of the tank and up to ground level. In this problem (and this problem only) it is permissible for you to use your calculator to evaluate any definite integrals that you need. 12 feet 4 feet 10 feet There is no gasoline in the vertical pipe when pumping begins The cylindrical tank is completely full of gas when pumping begins6. An apartment complex stores its salt in a shed that has a conical shape (see diagram below). The height of the shed is H meters and the radius of the base of the shed is R meters. At a vertical height of x meters from the ground, the density of the salt in units of kilograms per cubic meter is given by the function: ! "x( )= k # H $ x( ), where k is a positive constant. H R (a) Assuming that the shed is completely full of salt, find the total mass of the salt in the shed in units of kilograms. (b) Assuming that the shed is completely full of salt, find the vertical coordinate of the center of mass of the shed. (c) Suppose that the shed is filled by pouring salt into the shed through a small hole at the top. Salt is moved to the small hole by a horizontal conveyor belt. The shed starts out completely