419 3.7 Variation 49. Stefan-Boltzmann Law The Stefan-Boltzmann law says that the radiation of heat R from an object is directly proportional to the fourth power of the kelvin temperature of the object. For a certain object, R = 213.73 at room temperature 1293K2. Find R, to the nearest hundredth, if the temperature increases to 335 K. 50. Nuclear Bomb Detonation Suppose the effects of detonating a nuclear bomb will be felt over a distance from the point of detonation that is directly proportional to the cube root of the yield of the bomb. Suppose a 100-kiloton bomb has certain effects to a radius of 3 km from the point of detonation. Find, to the nearest tenth, the distance over which the effects would be felt for a 1500-kiloton bomb. 51. Malnutrition Measure A measure of malnutrition, called the pelidisi, varies directly as the cube root of a person’s weight in grams and inversely as the person’s sitting height in centimeters. A person with a pelidisi below 100 is considered undernourished, while a pelidisi greater than 100 indicates overfeeding. A person who weighs 48,820 g with a sitting height of 78.7 cm has a pelidisi of 100. Find the pelidisi (to the nearest whole number) of a person whose weight is 54,430 g and whose sitting height is 88.9 cm. Is this individual undernourished or overfed? Weight: 48,820 g Weight: 54,430 g 78.7 cm 88.9 cm 52. Photography In photography, the f-stop number, N, is directly proportional to the focal length of the lens, ƒ, and inversely proportional to the aperture diameter, D. N = ƒ D The aperture is a roughly circular opening of the lens that determines the amount of light admitted to the camera sensor. Because the focal length and aperture diameter are measured in the same units (millimeters), the f-stop number is dimensionless. (a) For an f-stop number of 2.5 and focal length 70 mm, what is the aperture diameter? (b) If the f-stop number is increased to 5 and the focal length remains the same, 70 mm, what is the aperture diameter? Concept Check Answer each question. 53. What happens to y if y varies inversely as x, and x is doubled? 54. What happens to y if y varies directly as x, and x is halved? 55. Suppose y is directly proportional to x, and x is replaced by 1 3x. What happens to y? 56. Suppose y is inversely proportional to x, and x is tripled. What happens to y? 57. Suppose p varies directly as r3 and inversely as t2. If r is halved and t is doubled, what happens to p? 58. Suppose m varies directly as p2 and q4. If p doubles and q triples, what happens to m?
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