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Distance Traveled Upon takeoff, the velocity readings of a rocket noted every second for
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Calculus & Its Applications (14th Edition)
- 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_forwardCost for a Can This is a continuation of Exercises 12 and 13. Suppose now that we use different materials in making different parts of the can. The material for the side of the can costs 0.10 per square inch, and the material for both the top and bottom costs 0.05 per square inch. a. Use a formula to express the cost C, in dollars, of the material for the can as a function of the radius r. b. What radius should you use to make the least expensive can? An Aluminum Can The cost of making a can is determined by how much aluminum A, in square inches, is needed to make it. This in turn depends on the radius r and the height h of the can, both measured in inches. You will need some basic facts about cans. See Figure 2.107. The surface of a can may be modeled as consisting of three parts: two circles of radius r and the surface of a cylinder of radius r and height h. The area of each circle is r2, In what follows, we assume that the can must hold 15 cubic inches, and we will look at various cans holding the same volume. a. Explain why the height of any can that holds a volume of 15 cubic inches is given by h=15r2 b. Make a graph of the height h as a function of r, and explain what the graph is showing. c. Is there a value of r that gives the least height h? Explain. d. If A is the amount of aluminum needed to make the can, explain why A=2r2+2rh. e. Using the formula for h from part a, explain why we may also write A as A=2r2+30r. An Aluminum Can, Continued This is a continuation of Exercise 12. The cost of making a can is determined by how much aluminum A, in square inches, is needed to make it. As we saw in Exercise 10, we can express both the height h and the amount of aluminum A in terms of the radius r: h=15r2A=2r2+30r a. What is the height, and how much aluminum is needed to make the can, if the radius is 1 inch? This is a tall, thin can. b. What is the height, and how much aluminum is needed to make the can, if the radius is 5 inches? This is a short, fat can. c. The first two parts of this problem are designed to illustrate that for an aluminum can, different surface areas can enclose the same volume of 15 cubic inches. i. Make a graph of A versus r and explain what the graph is showing. ii. What radius should you use to make the can using the least amount of aluminum? iii. What is the height of the can that uses the least amount of aluminum?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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