Concept explainers
For Problems 10 to 14, use a computer as needed to make plots of the given surfaces and the isothermal or equipotential curves. Try both
(a) Suppose that a hill (as in Fig. 5.1 ) has the equation
(b) If you start at the point
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Mathematical Methods in the Physical Sciences
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- Graph the level curves in the first quadrant of the xy-plane for the following functions at heights of z = 0, z = 2, and z = 4; 3x + y + 2z = 8arrow_forwardHomework #5 Find the areas enclosed by the following curves: (а) у %3D х2 — 2 and y = 2 - %3D (b) y = x* – 4x² + 4 and y = x² %Darrow_forwardWhat do I put into R studio to get this information? How does the size of the diamond relate to the price? Make a series of fitted curves that illustrate the relationship between price of a diamond and its carat weight based upon its clarity.arrow_forward
- 1. Find the equation of the line whose y-intercept is -7 and whose slope 4/3. 2. Find the equation to the line through the origin and with an inclination of arcton -5/7.arrow_forwardA trough 5 feet long and 1 foot high is filled with water. The vertical cross-sections of the trough parallel to the ends are shaped like the graph of y=x¹ for x = -1 to z = 1. The amount of work required to empty the trough by pumping the water over the top edge is: foot-poundsarrow_forwardAn airport can be cleared of fog by heating the air. The amount of heat required depends on the air temperature and the wetness of the fog. The figure below shows the heat H(T,w) required (in calories per cubic meter of fog) as a function of the temperature T (in degrees Celsius) and the water content w (in grams per cubic meter of fog). Note that this figure is not a contour diagram, but shows cross-sections of H with w fixed at 0.1, 0.2, 0.3, and 0.4. (a) Estimate HT (10,0.2): HT (10,0.2) ≈ (Be sure you can interpret this partial derivative in practical terms.) (b) Make a table of values for H(T,w) from the figure, and use it to estimate HT (T,w) for each of the following: T = 20,w = 0.2 : HT (T,w) ≈ T = 30,w = 0.2 : HT (T,w) ≈ T = 20,w = 0.3 : HT (T,w) ≈ T = 30,w = 0.3 : HT (T,w) ≈ (c) Repeat (b) to find Hw(T,w) for each of the following: T = 20,w = 0.2 : Hw(T,w) ≈ T = 30,w = 0.2 : Hw(T,w) ≈ T = 20,w = 0.3 : Hw(T,w) ≈ T = 30,w = 0.3 : Hw(T,w) ≈ (Be sure you can interpret this…arrow_forward
- An airport can be cleared of fog by heating the air. The amount of heat required depends on the air temperature and the wetness of the fog. The figure below shows the heat H(T, w) required (in calories per cubic meter of fog) as a function of the temperature T (in degrees Celsius) and the water content w (in grams per cubic meter of fog). Note that this figure is not a contour diagram, but shows cross-sections of H with w fixed at 0.1, 0.2, 0.3, and 0.4. H(T,8.4) 500 H(T,8.3) B00 H(T,8.2) 200 100 H(T,8.1) (a) Estimate HT(10,0.2): HT(10,0.2) (Be sure you can interpret this partial derivative in practical terms.) (b) Make a table of values for H(T, w) from the figure, and use it to estimate HT(T, w) for each of the following: Т%3 20, w — 0.2: Нт(Т, w) T = 30, w = 0.2 : HT(T, w) = T = 20, w = 0.3 : HT(T,w) z T = 30, w = 0.3 : HT(T, w) 2arrow_forwardIf you were to reflect point A across the Y = x line, what would the coordinates of the new point be?arrow_forwardSketch a contour map for the graph of the equation z = x² + y². Include level curves corresponding to z = - 2, z = - 1, z = 0, z = 1, and z = 2, if possible.arrow_forward
- Consider an axisymmetric feeding bowl as shown on the left below. Its cross-section is shown on the right with the axis of symmetry aligned along the y-axis. The top curve of the cross-section is given by the function b. What is the capacity of this bowl? Capacity= cm³ c. Find the volume of water inside the bowl if the water level is 3.3 cm measured from the bottom. Volume of water = 1 961 y = C 961 The horizontal base joins the endpoints of the top curve. Both x and y are in cm. You may need to use a computer algebra system to solve some of the following problems. Round all your answers to at least 3 significant figures. a. Assume that the bowl is solid and is made with plastic. Find the amount of plastic (as a volume) required to make this bowl. Amount of plastic = cm³ cm3 -x4 (9-x) 1 4 -x¹ (9 + x) y 0≤x≤9.1 -9.1 < x < 0 Xarrow_forwardThe point L(5, -4) is reflected over the line y = –5. What are the coordinates of the resulting point, L′?arrow_forwardWhile it is often stated that the boiling point of water is 100°C (212°F), in reality water boils at different temperatures at different altitudes. Gerald collected data on the boiling point of water at different altitudes. His findings are as follows: Construct a scatter plot for this data on the graph. Use 1 centimeter on the horizontal axis to represent 1,000 feet. Use 1 centimeter on the vertical axis to represent 2°F from the interval of 180°F to 220°F. Sketch a line that appears to best fit the data and label a point on the graph for 3,200 feet.arrow_forward
- Trigonometry (MindTap Course List)TrigonometryISBN:9781305652224Author:Charles P. McKeague, Mark D. TurnerPublisher:Cengage Learning