3.4 Quadratic Modeling.

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Presentation transcript:

3.4 Quadratic Modeling

Quiz L: length W: width H: height Write out the formulas for the area of a rectangle, A, and Volume for a box, V. A = _____ V = _____

Height of a Projected Object If air resistance is neglected, the height s ( in feet ) of an object projected directly upward from an initial height s0 feet with initial velocity v0 feet per second is s (t) = -16t2 + v0t + s0, where t is the number of seconds after the object is projected.

Application A ball is thrown directly upward from an initial height of 100 feet with an initial velocity of 80 feet per second. Give the function that describes the height of the ball in terms of time t. Graph this function so that the y-intercept, the positive x-intercept, and the vertex are visible. If the point (4.8, 115.36) lies on the graph of the function. What does this mean for this particular situation? After how many seconds does the projectile reach its maximum height? What is the maximum height? Solve analytically and graphically. For what interval of time is the height of the ball greater than 160 feet? Determine the answer graphically. After how many seconds will the ball fall to the ground? Determine the answer graphically.

Finding the Area of a Rectangular Region A farmer wishes to enclose a rectangular region. He has 120 feet of fencing and plans to use his barn as one side of the enclosure. Let x represent the length of one of the parallel sides of the fencing. a) Determine a function A that represents the area of the region in terms of x. b) What are the restrictions on x? c) Graph the function in a viewing window that shows both x-intercepts and the vertex of the graph. d) shows the cursor at (18, 1512). Interpret this information. e) What is the maximum area the farmer can enclose?

Finding the Volume of a Box A machine produces rectangular sheets of metal satisfying the condition that the length is three times the width. Furthermore, equal-sized squares measuring 5 inches on a side can be cut from the corners so that the resulting piece of metal can be shaped into an open box by folding up the flaps. a) Determine a function V that expresses the volume of the box in terms of the width x of the original sheet of metal b) What restrictions must be placed on x? c) If specifications call for the volume of such a box to be 1435 cubic inches, what should the dimensions of the original piece of the metal be? d) What dimensions of the original piece of metal will assure a volume greater than 2000, but less than 3000, cubic inches? Solve graphically.

Homework PG. 198: 10, 15, 22, 31, 32, 37, 40, 41 KEY: 15, 22, 37 Print out project 2 and bring it to class next time. We are going to finish project 2 together and turn it in at the end of the class.