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Unformatted text preview: 2.4 Functions as Real World Models Many of the processes studied in the physical and social sciences involves understanding how one quantity is related to another quantity. Finding the function that describes the dependence of one quantity to another is called modeling. Modeling real world problems especially those that require optimization is one of the important applications of the study of functions. Let us consider now some examples. Example 2.4.1 Emma and Brandon drive away from a campground at right angles to each other. Emma’s speed is 65 kph and Brandon’s is 55 kph. a) Express the distance between the cars as a function of time. b) Find the domain of the function. Solution: a).After 1 hour, Emma has traveled 65 km and Brandon has traveled 55 km. Since they drove at right angles to each other, we can use the Pythagorean theorem to find the distance between them. This distance would be the length of the hypotenuse of a triangle with legs measuring 65 km and 55 km. After 2 hours, the triangle’s legs would measure 130 km and 110 km. Observe that the distances covered by each person will depend on the time that has passed and the speed of the person. Here we use the fact that distance = speed x time. After t hours, Emma and Brandt have traveled 65 t km and 55 t km, respectively. Using the Pythagorean theorem: 𝑑 ( 𝑡 ) 2 = (65 𝑡 ) 2 + (55 𝑡 ) 2 Because distance must be nonnegative, we need consider only the positive square root when solving for d ( t ): 𝑑 𝑡¡ = ¢ 65 𝑡¡ 2 + 55 𝑡¡ 2 Thus, 𝑑 𝑡¡ = 85.15 𝑡 , 𝑡 ≥ . b) Since the time traveled, t , must be nonnegative, the domain is the set of nonnegative real numbers [0, + ∞ ) . Example 2.4.2 A rectangular field is to be fenced along the bank of a river, and no fence is required along the river. The material for the fence costs PhP8 per running foot for the two ends and PhP12 for running foot for the side parallel to the river; PhP3600 worth of fence is to be used. a. Let x be the length of an end; express the number of square feet in the area of the field as a function of x b. What is the domain of the resulting function? Solution a) Let y be the length of the side of the field parallel to the river and A square feet be the area of the field. Then 𝐴 = ?? and because the cost of the material for each end is PhP8 per running foot and the length of an end is x feet , the total cost of the fence for each end is 8 x pesos....
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This note was uploaded on 05/13/2011 for the course MATH 17 taught by Professor Dikopaalam during the Spring '11 term at University of the Philippines Los Baños.
- Spring '11