Find the Surface Area of a Prism

Transcription

1 Find the Surface Area of a Prism The of a solid is the area of the surface of the solid. A is a polyhedron with two parallel congruent bases. The bases are two faces of the polyhedron. The other faces are called the lateral faces. The (LA) of a prism is the sum of the area of the lateral faces. The (SA) of a prism is the sum of the lateral area and the area of the two bases. Find the Surface Area of the prism. SA = 2(19 X 29) + 2(29 X 6.5) + 2(19 X 6.5) SA = 2( ) + 2( ) + 2( ) SA = sq cm

2 Find the Surface Area of a Cylinder A is a solid that has parallel bases that are circles. The is a segment that joins the planes of the bases and is perpendicular to each. The length of the altitude is called the. For a cylinder Lateral Area (LA) = 2 rh. Surface Area (SA) = 2 r rh Find the Lateral Area of the cylinder to the nearest whole number. LA = 2 π rh LA = 2 π _ LA = π square cm (exactly) LA = (approximately) = sq cm

3 Find the Surface Area of the cylinder in terms of π. SA = 2 π r π rh SA = 2 π _ + 2 π _ _ SA = π + π sq cm SA = sq cm

4 Right Triangles Triangle Theorem In a triangle both legs are (equal measure) and the length of the hypotenuse is times the length of a leg. Find the value of each variable. If your answer is not an integer, Write it in simplest radical form with the denominator rationalized. x = 45 y y = 8 45 x A bridge has right triangular supports. If the hypotenuse of each support is 10 ft, find the length of each leg of a support. hyp = leg Rationalize the denominator ft 10 = x = = = ft x = x The approximate length is ft x

5 Right Triangles Triangle Theorem In a triangle the length of the hypotenuse is the length of the shorter leg and the length of the longer leg is the times the length of the shorter leg. hypotenuse = 2 shorter leg longer leg = 3 shorter leg Find the value of the variable. If your answer is not an integer, Write it in simplest radical form with the denominator rationalized. hyp = shorter leg x = 2 x = x

6 Find the value of each variable. If your answer is not an integer, Write it in simplest radical form with the denominator rationalized. Remember shorter leg is half of hypotenuse so x = longer leg = 3 shorter leg so 40 x y = 3 = 30 y

7 Trigonometric Ratios and Basic Trigonometry B C a A b c Sine of A = = Cosine of A = = Tangent of A = = Write the ratios for sin M, cos M, and tan M. Give the exact value and a four decimal place approximation. sin M = = M 9 7 K 4 cos M = = tan M = =

8 Practice finding trigonometric ratios with the calculator. Check to make sure your calculator is in degree Mode. Enter the problem into the calculator to be sure you get what is shown on the video. cos 58 = Find the value of x. Round to the nearest 10 th. tan = 7 64 x tan 64 = tan 64 = 7 tan 64 = x x A skateboarding ramp is 12 high and rises at an angle of 17. How long is the base of the ramp? Round to the nearest inch. tan = 17 x (base of ramp) 12 tan 17 = tan 17 = x tan 17 = 12 Divide both sides by tan 17 = x

9 Work with me Use a calculator to approximate the measure of acute angle A to the nearest whole degree. cos A = 0.2 Learn how your calculator finds cos 1 or inverse cosine. This will give you the angle whose cosine is 0.2 Then enter cos 1 (.2) So the measure if angle A is approximately 78 Work with me Use a calculator to approximate the measure of acute angle A to the nearest whole degree. sin A = sin 1 (.1736) So measure of angle A is approximately 10 Work with me Find the value of x. Round to the nearest degree. Which function uses? 3 x 5.8 cos 1 (. ) So x is approximately 59

10 Use Angles of Elevation and Depression to Solve Problems Angles of Elevation and Depression If the line of sight is upward from the horizontal, the angle is an angle of. (elevation or depression) If the line of sight is downward from the horizontal, the angle is an angle of. (elevation or depression) 5 is an angle of from Max to the top of the waterfall. is 7aiis an angle of from the top of the waterfall to Max.

11 A train track through a mountain is 2000 ft long and makes an angle of 1.5 with the horizontal. What is the change in elevation from one end of the tunnel to the other? 2000 ft 1.5 x Which trigonometric function relates to side opposite and hypotenuse? (sine, cos, or tan) sin 1.5 = = sin 1.5 = 2000 = x x ft Work with me There are over 3500 hot air balloons in the United States. Suppose a hot air balloon is flying at an altitude of 2100 ft. If the angle of depression from the pilot of the balloon to a house is 31, how far from the house is the pilot? 31 x 31 balloon 2100 ft sin 31 = = sin 31 = house x = 2100 x = ft

12 Define U.S. Units of Length and Convert Recall U.S. Units of Length 12 inches (in) = 1 foot (ft) 3 feet = 1 yard (yd) 36 inches = 1 yard 5280 feet = 1 mile (mi) A is a fraction that equals 1 Examples: 1 1 Convert 60 in. to feet ft Convert 8 to inches First change 8 to an improper fraction = in

13 Convert 10 ft to yards or yd

14 Define Metric Units of Length and Convert Metric Units of Length 1 kilometer (km) = 1000 meters (m) 1 hectometer (hm) = 100 m 1 dekameter (dam) = 10 m 1 meter (m) = 1m 1 decimeter (dm) = 1/10 m or 0.1 m 1 centimeter (cm) = 1/100 m or 0.01 m 1 millimeter (mm) = 1/1000 m or m Use this chart to help with conversions. km hm dam m dm cm mm Convert 1500 cm to meters: Convert 0.04 m to millimeters: 1500 cm = m 0.04 m = mm

15 Define U.S. Units of Weight and Convert Recall U.S. Units of Weight 16 ounces (oz) = 1 pound (lb) 2000 pounds = 1 ton To convert in the U.S. system we use. Examples: 1 1 Convert 60 oz to lb lb Convert 4.9 tons to pounds = lb

16 Convert 89 oz to a mixed unit of pounds and ounces So 89 oz = lb oz

17 Define Metric Units of Mass and Convert Metric Units of Mass 1 kilogram (kg) = 1000 grams (g) 1 hectogram (hg) = 100 g 1 dekagram (dag) = 10 g 1 gram (g) = 1g 1 decigram (dg) = 1/10 g or 0.1 g 1 centigram (cg) = 1/100 g or 0.01 g 1 milligram (mg) = 1/1000 g or g Use this chart to help with conversions. kg hg dag g dg cg mg Convert 4 g to milligrams: Convert 6.3 g to kilograms: 4 g = mg 6.3 g = kg

18 Define U.S. Units of Capacity and Convert Recall U.S. Units of Capacity 8 fluid ounces (fl oz) = 1 cup (c) 2 cups = 1 pint (pt) 2 pints = 1 quart (qt) 4 quarts = 1 gallon (gal) To convert in the U.S. system we use. Examples: 1 1 Convert 14 qt to gal gal Convert 42 cups to quarts = qt

19 Convert 58 quarts to a mixed unit of quarts and gallons So 58 qts = gal qt

20 Define Metric Units of Capacity and Convert Metric Units of Capacity 1 kiloliter (kl) = 1000 liters (L) 1 hectoliter (hl) = 100 L 1 dekaliter (dal) = 10 L 1 liter (L) = 1L 1 deciliter (dl) = 1/10 L or 0.1 L 1 centiliter (cl) = 1/100 L or 0.01 L 1 milliliter (ml) = 1/1000 L or L Use this chart to help with conversions. kl hl dal L dl cl ml Convert 5600 ml to liters: Convert 0.16 kl to liters: 5600 ml = L 0.6 kl = L

21 The Distance and Midpoint Formulas Distance Formula The distance d between two points, and, is given by Find the distance between ( 3, 2) and ( 1, 3) Midpoint Formula The midpoint of the line segment whose endpoints are, and, is the point with coordinates 2, 2 Find the midpoint: ( 2, 1) and ( 8, 6 ), = (, = (, )

22 Write Square Roots of Negative Numbers in the Form bi The Imaginary Unit The imaginary unit, written i, is the number whose square is 1. That is and = = i =

23 Complex Number A complex number is a number that can be written in the form a + bi where a and b are real numbers = 1 2 = =

24 Add or Subtract Complex Numbers Sum or Difference of Complex Numbers If a + bi and c + di are complex numbers then their sum is defined as follows: Their difference is defined as follows: (4 7i) + (2 + 3i) 4 7i + + ) 6

25 Multiply Complex Numbers ( 10i)( 4i) 40 (6 2i) (3 + i) ( ) 18 ( )

26 Vertical and Horizontal Shifts Common Graphs Use the graphs pictured here to assist with the following problems.

27 Vertical Shifts (Upward and Downward) Let k be a positive number Graph of Same as Moved g(x) =f(x) + k f(x) k units upward g(x) = f(x) k f(x) k units downward Sketch the graph of 3. Shift the graph of 3_ units (up or down) Sketch the graph: 2. (Refer to the graph of shown in the box on the previous page.) Shift the graph (up or down) 2 units.

28 Horizontal Shifts (Left and Right) Let k be a positive number Graph of Same as Moved g(x) =f (x k) f(x) k units to the right g(x) = f (x + k) f(x) k units to the left Sketch the graph Shift the graph of units to the (right or left) ( 3, 4)

29 Sketch the graph Shift the graph of units to the (right or left) Sketch the graph: 11 Shift the graph of units to the (right or left) AND units (up or down)

30 Sketch the graph of 3 1. Shift the graph of AND units to the (right or left) units (up or down) Sketch the graph Shift the graph of AND units to the (right or left) units (up or down)

31 Reflect Graphs Reflection about the x axis The graph of g(x) = f(x) is the graph of f(x) reflected about the x axis. Sketch the graph The graph of shifts units to the (right or left) AND opens (upward or downward)

32 Sketch the graph of 2 3 The graph of shifts units to the (right or left) AND units to the (up or down) AND opens (upward or downward)

33 Stretching and Compressing Graphs of Absolute Value Functions Sketch the graph The vertex is at (, ) and the v is open (upward or downward)

34 Sketch the graph: (shown in red) Sketch the graph : (shown in red)

35 The Graph of the Absolute Value Function The graph of ( f(x) = a ) has vertex ( h, k) and is V-shaped opens up is a > 0 (pos) and down if a < 0 (neg) If < 1, the graph is wider than the graph of If > 1, the graph is narrower than the graph of Sketch the graph The vertex of the V is at (, ) The V opens (upward or downward) The graph is (wider or narrower) than ( 3, 4)

36 Graphing Piecewise Functions Sketch the graph x > 1 x < 1 x y x y The Domain of the function is The Range of the function is

37 Solve a Non linear System by Substitution A nonlinear system of equations: a system of equations with at least one nonlinear equation Choose either equation and solve for either variable. (Try to pick the simplest one.) Choosing the second equation, solve for x. x 2y = 4 x = Substitute the expression you got for x into the other equation. y 2 = 4 ( ) y 2 = So we have a quadratic equation. Set the right side = 0. y 2 + 2y = 0 y ( ) = 0 y = 0 or y + 2 = 0 y = 0 or y = This means there are two solution points and we know the y coordinates of each point. Substitute the values you got for y back into the equation to find x. x = 2y + 4 x = 2(0) + 4 = x = 2( 2) + 4 = So the solutions are (4, 0) and (, 2 )

38 Solve a Non linear System by Elimination Addition Method: Multiply one or both equations by a nonzero number so that the coefficients of one variable become opposites. Note that multiplying the second equation by 2 will create opposite coefficients for x 2 So now Adding these two equations gives: Divide both sides by 5 to get y 2 = y 2 = So there are two choices for y, 2 or 2. Substitute these back into original equation to find x x 2 = x 2 = x =

39 This means when y = 2, x can be either 1 or 1. This gives us 2 solutions ( 1, 2 ) and ( 1, 2 ) Recall there were two possible solutions for y, 2 and 2. We substituted 2 and found two solutions for x. We now need to go back and substitute ( 2) for y. This will give a very similar result x 2 = x 2 = x = So we have two additional solutions: ( 1, 2 ) and ( 1, 2 ) Thus there are four solutions to this system of nonlinear equations. (, ), (, ), (, ) and (, )

40 Addition Rule of Probability A compound event is any event combining two or more simple events. Addition Rule P (A or B) = P (the event A occurs or the event B occurs or they both occur) P (A or B) = P(A) + P(B) P(A and B) Mutually Exclusive Events A and B are disjoint (or ) if they cannot both occur together. Determine whether the events are disjoint (mutually exclusive.) a) Randomly selecting a cardiac surgeon; randomly selecting a female physician These two events could both occur at the same time. You could select a female cardiac surgeon. Therefore the events are (disjoint or not disjoint) b) Randomly selecting someone treated with the cholesterol reducing drug Lipitor; randomly selecting someone in a control group given no medication. A person could not be both taking the medication and not taking the medication so these events could not occur at the same time. Therefore the events are (disjoint or not disjoint)

41 Addition Rule If two events are disjoint (or mutually exclusive), P (A or B) = P(A) + P(B) Consider the following table which summarizes the results from the sinking of the Titantic Men Women Boys Girls Survived Died a) If one of the passengers of the Titantic is randomly selected find the probability of getting someone who is a woman or a child. First add to find the total number of people on the Titanic was. P (woman or child) = P (woman) + P (child) b) Find the probability of getting someone who is a child or someone who survived the sinking. Events are not disjoint since a person could be both a child and a survivor. Thus, P (child or survivor) = P (child) + P (survivor) P (child and survivor)

42 Complement The,, of a set A consists of the outcomes in which A does not occur. P( A or ) = P(A) + P ( ) = 1 This implies P(A) = 1 P ( ) and P ( ) = 1 P(A) If P(A) =.05, find P ( ) P ( ) = 1 P(A) P ( ) = 1.05 = Based on data from the Census Bureau, when a woman over the age of 25 is randomly selected, there is a probability that she has a bachelor s degree. If a woman over the age of 25 is randomly selected, find the probability that she does not have a bachelor s degree. P(no bachelor s) = 1 P(bachelor s) P ( ) = 1 P(A) P ( ) = =

43 Multiplication Rule of Probability: Basics Notation P (A and B) = P (the event A occurs in a first trial and event B occurs in a second trial) Conditional Probability: P (B/A) represents the probability of event B occurring after it is assumed that event A has already occurred. Independent Events Two events A and B are if the occurrence of one does not affect the probability of the occurrence of the other. If A and B are not independent, they are said to be. Determine whether the events are independent or dependent. a) Randomly selecting a quarter made before the year 2001 Randomly selecting a second quarter made before the year 2001 Answer: The date on the first quarter does not change the probability of the date on the second quarter. Therefore the events are. b) Randomly selecting a TV viewer who is watching The Barry Manilow Biography Randomly selecting a second TV viewer who is watching The Barry Manilow Biography Answer: What the first viewer is watching does not change what the second viewer is watching. So the events are.

44 c) Wearing plaid shorts with black socks and sandals Asking someone on a date and getting a positive response Answer: The person who is asked on the date might make a choice based on what the asker is wearing, if they do or do not like the plaid shorts with black socks and sandals. So these events are. Multiplication Rule P( A and B ) = P(A) * P (B/A) A biologist works with a sample of two vascular plants (denoted by V) and four nonvascular plants (denoted by N.) Listed below are the codes for the six plants being studied. She wants to randomly select two of the plants for further experimentation. Find the probability that the first selected plant is nonvascular and the second plant is also nonvascular. Assume that the selections are made (a) with replacement; and (b) without replacement. a) With replacement V V N N N N P (nonvascular) = Since the question says with replacement we are putting the first plant back before selecting the second so the probability for the second event is also. Use the multiplication rule to get P (A and B) = = =

45 b) Without replacement P (nonvascular) = Since the question says without replacement we are not putting the first plant back before selecting the second so the probability for the second event changes. P (nonvascular given that one nonvascular has been removed) = Use the multiplication rule to get P (A and B) = = = Recent developments appear to make it possible for couples to dramatically increase the likelihood that they will conceive a child with the gender of their choice. In a test of a gender selectiion method, 12 couples try to have baby girls. If this gender selection method has no effect, what is the probability that the babies will all be girls? P (all girls) = P (female) * P (female) * * P (female) [ 12 of these] = 0.5 * * * = = or as a fraction If there are actually 12 girls among 12 children, does the gender selection method appear to be effective? Why?

46 Multiplication Rule: Complements and Conditional Probability Find the probability of a couple having at least one girl among three children. Assume that boys and girls are equally likely and that the gender of a child is independent of the gender of any brothers or sisters. Step 1: Use a symbol to represent the event desired. A = Step 2: Identify the event that is the complement of A. = not getting at least one girl among three children = Step 3: Find the probability of the complement P ( ) = P ( boy and boy and boy ) = Step 4: Find P(A) by evaluating 1 P ( ): P(A) = 1 = Probability of At Least One To find the probability of at least one of something, calculate the probability of none, then subtract that result from 1. That is, P (at least one) = 1 P (none)

47 Conditional Probability A of an event is a probability obtained with the additional information that some other event has already occurred. P(B/A) denotes the conditional probability of event occurring given that event A has already occurred. It can be found by dividing the probability of events A and B both occurring by the probability of event A: Intuitive Approach to Conditional Probability The conditional probability of B given A can be found by assuming that event A has already occurred and, working under that assumption, calculating the probability the event B will occur. Republican Democrat Independent Male Female If we randomly select one senator, what is the probability of getting a Republican given that a male was selected? Males = + + = 86 P (Republican given Male) = P (B/A) = = 0.535

48 Solve Polynomial Inequalities of Degree 2 or Greater Work this quadratic inequality with me First look at the related equation: The solutions to this related equation are:, Place the solutions to the related equation on a number line: Pick a test point for the region on the left: Check if the test point make the original inequality a true of false statement: True or False? Check the other two regions: In interval notation, write down the regions that are solutions regions:

49 Solving a Polynomial Inequality 1. Write the inequality in standard from and then solve the related equation. 2. Separate the number line into regions with the solutions from Step For each region, choose a test point and determine whether its value satisfies the original inequality. 4. The solution set includes the regions whose test point value is a solution. If the inequality symbol is or, the values from Step 1 are solutions; if < or >, they are not. Work this quadratic inequality with me:

50 Work this polynomial inequality with me: