Warm Up. Wages = hourly rate hours worked. 1. Graph wages as a function of hours worked for r = $4, $10, $20.

Transcription

1 Warm Up Wages = hourly rate hours worked W r, t = rt 1. Graph wages as a function of hours worked for r = $4, $10, $0.. Graph hours worked as a function of hourly rate for W = $40, $100, $00.

2 Warm Up W r, t = rt Wages = hourly rate hours worked 1. Graph wages as a function of hours worked for r = $4, $10, $0.

3 Warm Up W r, t = rt Wages = hourly rate hours worked. Graph hours worked as a function of hourly rate for W = $40, $100, $00.

4 4.7 FORMING FUNCTIONS FROM VERBAL DESCRIPTIONS Objectives: 1. Form a function from a verbal description.. Determine the maximum and minimum values.

5 An important concern of mathematics is finding the minimum or maximum value of a function. This can help you minimize or maximize profits, minimize stress on a girder or maximize the volume of a container made from a given amount of material. Minimum and maximum vales are often referred to as extreme values. Approximate extreme values of a function can be found using a calculator. Exact extreme values are most often found using calculus. Whether technology or calculus is used, we always need to write a rule for the function to be minimized or maximized. If the rule depends on two or more variables, then we also need to find a relationship among the variables so the function can be written in terms of only one variable. Developing these skills is the goal of today s lesson.

6 Express the area A of a circle as a function of its circumference C C = πr so r = C π A = πr = π C π = C 4π A(C) = C 4π Express the circumference C of a circle as a function of its area A C = 4πA so C = 4πA = πa C(A) = πa

7 A cone is inscribed in a sphere with radius 1, as shown in the diagram. Express the height of the cone in terms of x. Express the radius of the cone in terms of x. r = Express the volume V of the cone in terms of x. h = x x V(x) = 1 3 π 144 x x + 1 What is the domain of x? 0 < x < 1 x

8 Work on the problem below with your group. An open box with a square base is to be constructed from sheet metal in such a way that the completed box is made of m of sheet metal. Express the volume of the box as a function of the base width.

9 4.7 Forming Functions from Verbal Descriptions An open top box with a square base is to be constructed from sheet metal in such a way that the completed box is made from m of sheet metal. Express the volume of the box as a function of base width. Surface V( w, h) w h A w b Write hin terms of w. 4 A s 4wh 4wh h w 4w w V w w V ( w) w 4w w 4 w 3 w w h

10 V( w) Graph V w 4 w 3 h w 4w on an appropriate window and find the width and height that maximizes the volume.

11 The width of the square base is 0.8 meters. The maximum volume is 0. 7 m 3

12 The width of the square base is 0.8 meters. The maximum volume is 0. 7 m 3 h w h w 4w units meters

13 4.7 Forming Functions from Verbal Descriptions A north-south bridal path intersects an east-west river at point O. At noon, a horse and rider leave O traveling north at 1 km/h. At the same time, a boat is 5 km east of O traveling west at 16 km/h. Express the distance d between the horse and the boat as a function of the time t in hours after noon.

14 h1t b5 16t By Pythagorean THM: d h b d 1t 5 16t d( t) 1t 5 16t 144t t 56t 400t 800t 65 t-coordinate of vertex? t By hand, determine what the minimum distance is between the horse and the boat. b a km

15 Classwork/Homework Page 161 #5,7,9,13,15,17

16 5) A store owner bought n dozen toy boats at a cost of $3.00 per dozen, and sold them at $.75 each. Express the profit P in dollars as a function of n. Profit = Income Cost P I C Let x = the number of boats sold I.75 x or $9.00 per dozen ( n) 9n C 3n P( n) I C P( n) 9n 3n P( n) 6n

17 9) A light 3 m above the ground causes a boy 1.8 m tall to cast a shadow s meters long. Express s as a function of d, the boy's distance in meters from the light. 3s 1.8s 1.8d 3 m 1.8 m 3s 1.8s 1.8d 1.s 1.8d d Similar Triangles: s s d s 1.8d s 1. s( d) 1.5d

18 13) A stone is thrown into a lake, and t seconds after the splash, the diameter of the circle of ripples is t meters. a) Express the circumference C of this circle as a function of t. t C() t t b) Express the area A of this circle as a function of. t At () t t 4

19 17) At :00 P.M. a bike A is 4 km north of point C and traveling south at 16 km / h. At the same time, bike B is km east of C and traveling east at 1 km / h. 4 16t C 1t c) What is the distance between the bikes when they are closest? d t t t ( ) d(.1) 400(.1) 80(.1) 0 4 km

20 1) P( x, y) is an arbitrary point on the line x y 10. (0,0) P( x, y) a) Express the distance d from the origin to P as a function of the x-coordinate of P. d x x y y 1 1 x 0 y 0 x y y x10 y x x d x x x d x x x ( )

21 1) P( x, y) is an arbitrary point on the line x y 10. b) What are the domain and range of this function? d x x x ( ) x 40x100 0 Domain b 4ac 0 Range b x a 4 d (4) 5(4) 40(4) Since 5 is the min, dx ( ) 5

22 5) From a raft 50 m offshore, a lifeguard wants to swim to shore and run to a snack bar 100 m down the beach. a) If the lifeguard swims at 1 m/s and runs at 3 m/s, express the total swimming and running time t as a function of the distance x shown in the diagram. Work

23 Write in terms of x. d x 50 d x x d rt t d / r tx ( ) x x x 3 x Problem

24 1 t( x) x x,0 x b) Use a calculator to find the minimum time seconds