Section 3.4. from Verbal Descriptions and from Data. Quick Review. Quick Review Solutions. Functions. Solve the equation or inequality. x. 1.

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1 .4: Quadratic Models Starter.4 Section.4 Building Quadratic at Models from Verbal Descriptions and from Data DO:,, p. 60 Solve the given formula for the required variable. ) Area of a Triangle Solve for b : A bh ) Volume of a Right Circular Cylinder Solve for h : V r h 4 ) Volume of a Sphere Solve for r : V r 4) Surface Area of a Sphere Solve for r : A 4 r 5) Simple Interest Solve for P : I Prt Starter.4 Solve the given formula for the required variable. A ) Area of a Triangle Solve for b: A bh b h V ) Volume of a Right Circular Cylinder Solve for h: V rh h r 4 V ) Volume of a Sphere Solve for r: V r r 4 A 4) Surface Area of a Sphere Solve for r: A 4 r r 4 I 5) Simple Interest Solve for P : I Pt r P rt Solve the equation or inequality. x. 9 0 x. 6 0 Quick Review Find all values of x algebraically for which the algebraic expression is not defined.. x 4. x x 5. x Quick Review Solutions Functions Solve the equation or inequality.. x 9 0. x 6 0 x x 4 Find all values of x algebraically for which the algebraic expression is not defined.. x x 4. x x 5. x x x

2 .4: Quadratic Models Functions (cont d) OBJECTIVE Revenue Building Quadratic Models amount of money received from sales of an item R = xp, where: x = units sold, p = unit selling price,000 x, p DO: # DO: #7 (a)find the maximum height of the projectile. (b)how far from the base of the cliff will the projectile strike the water? DO: #

3 Precalculus.4: Quadratic Models (a) Write the volume V as a function of x. (b) Find the domain of V as a function of x. (c) Graph V as a function of x over the domain found in part (b) and use the maximum finder on your grapher to determine the maximum volume such a box can hold. (d) How big should the cut-out squares be in order to produce the box of maximum volume? DO: # (a) Write the volume V as a function of x. (a) Write the volume V as a function of x. (a) The width 8 x and the length 5 x. The depth is x when the sides are folded up. V x 8 x 5 x (b) Find the domain of V as a function of x. (b) Find the domain of V as a function of x. (b) The depth of x must be nonnegative, as must the side length and width. The domain is [0,4] where the endpoints give a box with no volume.

4 Precalculus.4: Quadratic Models (c) Graph V as a function of x over the domain found in part (b) and use the maximum finder on your grapher to determine the maximum volume such a box can hold. (c) Graph V as a function of x over the domain found in part (b) and use the maximum finder on your grapher to determine the maximum volume such a box can hold. The maximum occurs at the point (5/, 90.74). The maximum volume is abo ut in. (d) How big should the cut-out squares be in order to produce the box of maximum volume? (d) How big should the cut-out squares be in order to produce the box of maximum volume? (d) Each square should have sides of one-and-two thirds inches. Applications of Quadratic Functions and Models Example A farmer wishes to enclose a rectangular region. He has 0 feet of fencing, and plans to use one side of his barn as part of the enclosure. Let x represent the length of one side of the fencing. (a) Find 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. (a) What is the maximum area the farmer can enclose? Solution (a) Area = width length, so A( x) x(0 x) (b) A( x) x 0x Since x represents length, x > 0. Also, 0 x > 0, or x < 60. Putting these restrictions together gives 0 < x < 60. Applications of Quadratic Functions and Models (c) Figure 8 pg - 60a (d) Maximum value occurs at the vertex. A( x) x 0x, where a, b 0. b 0 x 0 a ( ) A(0) (0) 0(0) 800 square feet 4

5 .4: Quadratic Models Finding the Volume of a Box Example A machine produces rectangular sheets of metal satisfying the condition that the length is times the width. Furthermore, equal size 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 45 cubic inches, what should the dimensions of the original piece of metal be? (d) What dimensions of the original piece of metal will assure a volume greater than 000 but less than 000 cubic inches? Solve graphically. (a) (b) (c) Finding the Volume of a Box Using the drawing, we have Volume = length width height, V ( x) (x 0)( x 0)(5) 5x 00x 500 Dimensions must be positive, so x 0 > 0 and x 0 > 0, or 0 x and x 0. Both conditions are satisfied when x > 0, so the theoretical domain is (0, ). 45 5x 00x x 00x 95 0 (5x 55)( x 7) x or x 7 Only 7 satisfies x > 0. The original dimensions : 7 inches by (7) = 5 inches. Finding the Volume of a Box Using Pythagorean Theorem (d) Set y = 5x 00x + 500, y = 000, and y = 000. Points of intersection are approximately (8.7, 000) and (.,000) for x > 0. Therefore, the dimensions should be between 8.7 and. inches, with the corresponding length being (8.7) 56. and (.) 6.6 inches. The longer leg of a right triangular lot is approximately 0 meters longer than twice the length of the shorter leg. The hypotenuse is approximately 0 meters longer than the length of the longer leg. Estimate the lengths of the sides of the triangular lot. Analytic Solution Let s = the length of the shorter leg in meters. Then s + 0 is the length of the longer leg, and (s + 0) + 0 = s + 0 is the length of the hypotenuse. s (s 0) (s 0) s 4s 80s 400 4s 0s 900 s 40s ( s 50)( s 0) 0 s 50 or s 0 The approximate lengths of the sides are 50 m, 0 m, and 0 m. Using Pythagorean Theorem Graphing Calculator Solution Replace s with x and find the x-intercepts of the graph of y y = 0 where y = x + (x + 0) and y = (x + 0). We use the x-intercept method because the y-values are very large. OBJECTIVE Figure 44 pg -64a The x-intercept is 50, supporting the analytic solution. 5

6 .4: Quadratic Models Quadratic Models Quadratic Models The percent of Americans 65 and older for selected years is shown in the table. Year, x % 65 and older, y (a) Plot data, letting x = 0 correspond to 900. (b) Find a quadratic function, f (x) = a(x h) + k, that models the data by using the vertex (0,4.). (c) Graph f in the same viewing window as the data. (d) Use the quadratic regression feature of a graphing calculator to determine the quadratic function g that provides the best fit. (a) (b) L = x-list and L = y-list Substituting 0 for h and 4. for k, and choose the data point (040,0.6) that corresponds to the values x = 40, y = f (40) = 0.6, we can solve for a. 0.6 a(40 0) ,600a a f ( x) x 4. Quadratic Models (c) (d) Figure 46a pg - 67a This is a pretty good fit, especially for the later years. Note that choosing other second points would produce other models. 6