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QUADRATIC FUNCTIONS the ball to reach the ground. The equation then becomes 0 = –0.5(9.8)(3.84) 2 + v o (3.84) + 1.45, which has a solution of v o approximately equaling 18.4 meters per second. Substituting this value into the general function will also provide enough information to help you find the maximum height of your throw. The equation h = –0.5gt 2 + v o t + h o can be simplified to h = –0.5gt 2 + h o for objects in freefall because v o = 0 when an object is dropped. Therefore if you plan to bungee-jump 200 meters off of a 250-meter-high bridge, then you should expect to be dropping for about 6.4 seconds. This value comes from substituting for the variables and solving the equation 50 = –0.5(9.8)t 2 + 250. (Note that the ending position will be 50 meters above the ground, since the rope is only extending 200 meters.) This general equation could also be used to estimate heights and times for other objects that are released at high heights, such as the steep drops on some amusement park rides. Horizontal distance, such as the distance traveled after slamming on the brakes in a car, can also be modeled with a quadratic function. In an effort to reconstruct a traffic accident, a law office could use the function d = 0.02171v 2 +0.03576v −0.24529 to determine how far a car could travel in feet, d, when breaking, or how fast it was moving in feet per second, v, before it started braking. The law office might also consider the average reaction time of 1.5 seconds upon seeing a hazardous condition. So the total stopping distance, t, can be modeled with the equation t = 0.02171v 2 + 0.03576v − 0.24529 + 1.5v, which simplifies to t = 0.02171v 2 + 1.53576v − 0.24529. Area applications can also be modeled by quadratic functions, because area is represented in square units. For example, pizza prices depend on the amount of pizza received, which is examining its area. However, on a pizza menu, the sizes are revealed according to each pizza’s diameter. If a 12-inch pie costs $12, a misconception would be to think that the 16-inch one should cost $16. A func- tion to represent the price, p, of this type of pizza in terms of its diameter, d, is p = 0.106π( d 2 ) 2 , because it is a unit cost times the pizza’s area. The value 0.106 is the price per square inch of pizza in dollars, assuming that the 12-inch pie for $12 will have the same unit-cost value as any other size pizza. Therefore a 16- inch pizza should cost p = 0.106π( 16 2 ) 2 ≈ $21.31. The restaurant, however, may decide to give a financial incentive for customers to purchase larger pies and reduce this price to somewhere near $20. Devising and purchasing tin cans for food are applications of surface area that can be represented by a quadratic function. Since most tin cans are cylindri- cal, the surface area can be determined by finding the area of the rectangular lat- eral area and the sum of the two bases, as shown in the following figure. If the manufacturer determines the height of its cans to be 4 inches tall with a variable radius, then the amount of sheet metal in square inches, a, needed for each can would be a = 8πr + 2πr 2 , where r is the radius of the can in inches. If tin costs the manufacturer $0.003 per square inch, then the materials cost, c, to produce each case of twenty-four cans can be represented by the function c = (24)(0.003)a = (24)(0.003)(8πr + 2πr 2 ), which simplifies to c ≈ 1.81r+ Download 1.81 Mb. Do'stlaringiz bilan baham: |
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