Concept explainers
Temperature Consider a circular plate of radius 1 given by
(a) Sketch the isotherm
(b) Find the hottest and coldest points on the plate.
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Chapter 13 Solutions
Calculus: Early Transcendental Functions (MindTap Course List)
- Minimum Find the minimum value of x2+20/(x+1) on the horizontal span of 0 to 10.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_forwardMaximum Sales Growth This is a continuation of Exercise 10. In this exercise, we determine how the sales level that gives the maximum growth rate is related to the limit on sales. Assume, as above, that the constant of proportionality is 0.3, but now suppose that sales grow to a level of 4 thousand dollars in the limit. a. Write an equation that shows the proportionality relation for G. b. On the basis of the equation from part a, make a graph of G as a function of s. c. At what sales level is the growth rate as large as possible? d. Replace the limit of 4 thousand dollars with another number, and find at what sales level the growth rate is as large as possible. What is the relationship between the limit and the sales level that gives the largest growth rate? Does this relationship change if the proportionality constant is changed? e. Use your answers in part d to explain how to determine the limit if we are given sales data showing the sales up to a point where the growth rate begins to decrease.arrow_forward
- The number of research article in the prominent journal Physical Review that were written by researchers in Europe during 1983 - 2003 can be modeled by P(t) = 7.0/ (1 + 5.4(1.2)^−t) where t is time in years since 1983. The graphs of P, P', and P" are shown below. Determine, to the nearest whole number, the value of t for which the graph of P is concave up and where it is concave down, and locate any points of inflection. Explain your methods thoroughly, including how the inflection point of f can be determined with each of the three graphs, some more clearly than others. What does the point of inflection tell you about scientific articles?arrow_forwardMaximum product Find numbers x and y satisfying the equation3x + y = 12 such that the product of x and y is as large as possible.arrow_forwardExtreme values of parabolas Consider the functionf(x) = ax 2 + bx + c, with a ≠ 0. Explain geometrically whyf has exactly one absolute extreme value on (- ∞, ∞). Findthe critical point to determine the value of x at which f has anextreme valuearrow_forward
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