APPLICATIONS
72. Turning a Corner A steel pipe is being carried down a hall-way that is
(a) Show that the length of the pipe in the figure is modeled by the function
(b) Graph the function
(c) Find the minimum value of the function
(d) Explain why the value of
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Algebra and Trigonometry (MindTap Course List)
- Graphical Reasoning Use the formulas for the area of a circular sector and arc length given in Section 1.1. (a) For =0.8, write the area and arc length as functions of r. What is the domain of each function? Use a graphing utility to graph the functions. Use the graphs to determine which function changes more rapidly as r increases. Explain. (b) For r=10 centimeters, write the area and arc length as functions of . What is the domain of each function? Use the graphing utility to graph the functions.arrow_forwardThe Doppler Effect As a train moves toward an observer (see the figure), the pitch of its whistle sounds higher to the observer than it would if the train were at rest, because the crests Of the sound waves are closer together. 'This phenomenon is called the Doppler effect. The observed pitch P is a function of the speed v of the train and is given by P(v)=poSoSov where PO is the actual pitch of the whistle at the source and So = 332 rn/s is the speed of sound in air. Suppose that a train has a whistle pitched at Po = 440 Hz. the function y = P(v) using a graphing device. How can the vertical asymptote of this function be interpreted physically?arrow_forwardSimple Harmonic Motion A mass suspended from a spring oscillates in simple harmonic motion at a frequency of 4 cycles per second. The distance from the highest to the lowest point of the oscillation is 100 cm. Find an equation that describes the distance of the mass from its rest position as a function of time. Assume that the mass is at its lowest point when t=0.arrow_forward
- Photography A photographer takes a picture of a three-foot-tall painting hanging in an art gallery. The camera lens is 1 foot below the lower edge of the painting (see figure). The angle subtended by the camera lens x feet from the painting is given by =arctan3xx2+4,x0. (a) Use a graphing utility to graph as a function of x. (b) Use the graph to approximate the distance from the picture when is maximum. (c) Identify the asymptote of the graph and interpret its meaning in the context of the problem.arrow_forwardView from a Satellite The figures on the next page indicate that the higher the orbit of satellite, the more of the earth the satellite can see. Let ,s, and h be as in the figure, and assume that the earth is a sphere of radius 3960 mi. a Express the angle as a function of h. b Express the distance s as a function of . c Express the distance s as a function of h. Hint: Find the composition of the functions in parts a and b. d If the satellite is 100 mi above the earth, what is the distance s that it can see? e How high does the satellite have to be to see both Los Angeles and New York, 2450 mi apart?arrow_forwardOscillating Spring A mass attached to a spring oscillates upward and downward. The displacement of the mass from its equilibrium position after t seconds is given by the function d=3.5cos(2t). where d is measured in centimeters (Figure 13). a. Sketch the graph of this function for 0t5. b. What is the furthest distance of the mass from its equilibrium position? c. How long does it take for the mass to complete one oscillation?arrow_forward
- Hyperbolic sine Function The hyperbolic sine function is defined by sinh(x)=exex2 a Sketch the graph of this function using graphical addition as in Exercise 17. b Use the definition to show that sinh(x)=sinh(x) Hyperbolic Cosine Function The hyperbolic cosine function is defined by cosh(x)=ex+ex2 a Sketch the graphs of the functions y=13ex and y=12ex on the same axes, and use graphical addition see Section 2.7 to sketch the graph of y=cosh(x). b Use the definition to show that cosh(x)=cosh(x).arrow_forwardGeometry A trough for feeding cattle is 4 meters long and its cross sections are isosceles triangles with the two equal sides being 12 meter (see figure). The angle between the two sides is . (a) Write the volume of the trough as a function of /2. (b) Write the volume of the trough as a function of and determine the value of such that the volume is maximized.arrow_forwardRadius of a Shock Wave An explosion produces a spherical shock wave whose radius R expands rapidly. The rate of expansion depends on the energy E of the explosion and the elapsed time t since the explosion. For many explosions, the relation is approximated closely by R=4.16E0.2t0.4. Here R is the radius in centimeters, E is the energy in ergs, and t is the elapsed time in seconds. The relation is valid only for very brief periods of time, perhaps a second or so in duration. a. An explosion of 50 pounds of TNT produces an energy of about 1015 ergs. See Figure 2.71. How long is required for the shock wave to reach a point 40 meters 4000 centimeters away? b. A nuclear explosion releases much more energy than conventional explosions. A small nuclear device of yield 1 kiloton releases approximately 91020 ergs. How long would it take for the shock wave from such an explosion to reach a point 40 meters away? c. The shock wave from a certain explosion reaches a point 50 meters away in 1.2 seconds. How much energy was released by the explosion? The values of E in parts a and b may help you set an appropriate window. Note: In 1947, the government released film of the first nuclear explosion in 1945, but the yield of the explosion remained classified. Sir Geoffrey Taylor used the film to determine the rate of expansion of the shock wave and so was able to publish a scientific paper concluding correctly that the yield was in the 20-kiloton range.arrow_forward
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