Concept explainers
Allometric Equation Many relations in biology are expressed by power functions, known as allometric equations, of the form
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- Volume of a Rocket A rocket consists of a right circular cylinder of height 20 m surmounted by a cone whose height and diameter are equal radius is the same as that of the section. What should this radius be (rounded to two decimal places) if the total volume is to be 500/3m3?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_forwardGraphical Reasoning Use a graphing utility to confirm the solutions found in Example 6 in two different ways. (a) Graph both sides of the equation and find the x -coordinates of the points at which the graphs intersect. Left side : y=cosx+1 Right side : y=sinx (b) Graph the equation y=cosx+1sinx find the x-intercepts of the graph. (c) Do both methods produce the same x -values? Which method do you prefer? Explain.arrow_forward
- Air Temperature As dry air moves upward, it expand and, in so doing, cools at a rate of about 1°C for each 100-meter rise, up to about 12 km. (a) If the ground temperature is 20°C, write a formula for the temperature at height h. (b) What range of temperatures can be expected if an air plane lakes off and reaches a maximum height of 5 km?arrow_forwardSpeed of a Skidding Car Police use the formula s=30fd To estimate the speed s (in mi/h) at which a car is traveling if it skids d feet after the brakes are applied suddenly. The number f is the coefficient of friction of the road, which is a measured of the “slipperiness” of the road. The table gives some typical estimates for f . If a car skids 65 ft. on wet concrete, how fast was is moving when the brakes were applied? If a car is traveling at 50 mi/h, how far will it skid on wet tar?arrow_forwardBlood Flow As blood moves through a vein or an artery, its velocity v is greatest along the central axis and decreases as the distance r from the central axis increases see the figure. The formula that gives v as the function of r is called the law of laminar flow. For an artery with radius 0.5 cm, the relationship between v in cm/s and r in cm is given by the function v(r)=18,500(0.25r2)0r0.5 a Find v(0.1)andv(0.4). b What do your answers to part a tell you about the flow of blood in this artery? c Make a table of values of v(r)forr=0,0.1,0.2,0.3,0.4,0.5. d Find the net change in the velocity v as r changes from 0.1 cm to 0.5 cm.arrow_forward
- Distance A plane flying at an altitude of 7 miles above a radar antenna passes directly over the radar antenna (see figure). Let d be the ground distance from the antenna to the point directly under the plane and let x be the angle of elevation to the plane from the antenna. ( d is positive as the plane approaches the antenna.) Write d as a function of x and graph the function over the interval 0x.arrow_forwardLinear Velocity A propeller with radius 3.50 feet is rotating at 900 revolutions per minute. Find the linear velocity of the tip of the propeller. Give the exact value and an approximation to three significant digits.arrow_forward
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