TAKS MATH PREPARATION

Transcription

1 Formula Sheet TAKS MATH PREPARATION Formula Sheet cont d The sum of all angles inside a triangle is 180 o (a + b) 2 = a 2 + b 2 + 2ab (a b) 2 = a 2 + b 2 2ab Area has the unit meter 2 or centimeter 2 while the volume has meter or centimeter Speed = distance/time with unit (m/s) or (km/s) Acceleration = speed/time with unit (m/s 2 ) or (km/s 2 ) The probability of any event is less than or equal to 1. It cannot be greater than 1 or less than 0. Problems Question: Find the slope of the line 2y = 8x (A) /2 (B) 4 (C) 8 (D) Not here The general line equation is y = mx + b where m is the slope of the equation So we have 2y = 8x Divide by 2 on both sides We get y = 4x /2 Compare this to the general line equation. We see that m = 4 and b = /2 So slope is m = 4 1

2 Question: Trapezoid KMPR is similar to trapezoid LNQS. Which is closest to the perimeter of trapezoid LNQS? (A) 2 units (B) 1 units (C) 25 units (D) 20 units Using the property of similarity which states that ratio of sides are equal to each other, we can say that the ratio of sides MK and NL is equal to the ratio of sides KR and LS Therefore MK NL KR LS But MK = 121/8, KR = 11 and LS = 4 So we get NL = 5.5 Now the triangle SQX is a right triangle. So using Pythagorean theorem we have a 2 + b 2 = c = c 2 c = or approx. 2 units Question: The drawing below shows how squares can be joined at their vertices to form a right triangle. Which is closest to the area in square inches of the largest square? (A) 1914 in 2 (B) 2 in 2 (C) 210 in 2 (D) 107 in 2 It has been mentioned that there are squares. So this means that each side is of equal length Alsothetriangleinthecenterisaright triangle 2

3 So AB 2 = AB = 2.76 in Now S is a square. Therefore area is AB 2 = which is approx. to 107 in 2 NowwenamethesquaresasS1,S2andS with the latter being the unknown. For square S1, the side AB = in For square S2, the side BC = 29.5 in But ABC is a right triangle. So using Pythagoras theorem,wehavea 2 +b 2 =c 2 Question: The area of the shaded portion of the rectangle shown below is 440 square feet. How can the area of the unshaded portion of the rectangle be expressed in terms of x in square feet? (A) 440x 0 (B) (0 + x)440 (C) (0 440)x (D) 0x 440 Area of rectangle is length*width = 0x Area of shaded part = 440 ft 2 Area of shaded + area of unshaded = 0x Therefore area of unshaded = 0x Area of shaded part Area of unshaded part = (0x 440) ft 2 Question: The length of each leg of an isosceles triangle is 5 centimeters more than twice the length of the base. If the perimeter of this isosceles triangle is 95 centimeters, what is the length of the base? (A) 17 cm (B) 21 cm (C) 0cm (D) 9 cm Perimeter of a triangle is sum of all the sides or (a + b + c) We know that an isosceles triangle has the same length on 2 sides. Assume that b is the base. So a=c Given that the sides are 5 cm more than twice the length of the base

4 Thereforea=5+2b Perimeter = 2a + b = b = 95 Upon solving for b we get 17 cm Question: A triangle is inscribed in a square, as shown below. What is the area of the shaded triangle inscribed in the square? (A) (1/4)x 2 (B) (1/2)x 2 (C) (1/8)x 2 (D) (1/16)x 2 Area of the shaded triangle is 0.5 * base * height Base is given as 0.5 x. Since the triangle is inscribed inside a square, the height is also 0.5 x So plugging in the values of base and height we have 0.5 * 0.5 * 0.5 x 2 = (1/8) x 2 Question: The price for this year s season tickets to a city hockey team s games was reduced by 15% from last year s ticket price, x. As a result, there was a 22% increase in the number of season tickets sold this year. If a total of 4000 season tickets were sold last year and each season ticket is equally priced, which expression could be used to determine the total sales from this year s season tickets? (A) 4000( )( )x (B) 4000( )(1 0.15)x (C) 4000(1 0.22)(1 0.15)x (D) 4000(1 0.22)( )x There is a reduction of 15% in price from last year s price of x. This means the net price is 100% 15% or (1.15) Also a 22% increase in the ticket sales compared to last year. This means 100% + 22% or ( ) The total sales of last year s ticket = 4000 x We can deduce that 22% increase is in comparison to last year s ticket sales which is ( ) 4000 x But it has been mentioned that there is a 15% reduction on the sale prices on x. So it is (1.15) ( ) 4000 x Question: The graph shows the distance a certain motorbike can travel at a constant speed with respect to time. Which of the following best describes the meaning of the slope of the line representing this situation? (A) The motorbike travels at a speed of about 8 miles per hour. (B) The motorbike travels at a speed of about 2.5 miles per hour. (C) The motorbike travels at a speed of about 5 miles per hour. (D) The motorbike travels at a speed of about 10 miles per hour. 4

5 So the speed = distance / time Speed = 5 / 0.5 = 10 miles/hr We can infer from the graph that at (1/2) hour the distance covered is 5 miles Question: The table below shows the relationship between p, the number of cell phones a company produces, and d, the number of cell phones that are defective. Which equation can be used to describe this relationship? (A) d = 20p (B) d = 0.05p (C) d = (p 90)/2 (D) d = p 95 The easiest way to solve this is to plug in the values of p in the choices to get the answer rather than wasting time to figure out the relationship First consider the option d = 20p. Now check for p = 100, 200. When we substitute for p we get d = 2000, So eliminate this Next try the same for d = 0.05p. We get the values as 5, 10 Sincethismatches,wecanselectthischoicebut many times we have to try all the choices to get the correct answer Question: Which lists the functions of the form y=ax 2 in order from the widest to the narrowest graph? (A) y = x 2,y= x 2,y= 2 x 2,y=2x 2 1 (B) y = x 2 2,y= x 2,y=2x 2 7,y= x (C) y = x 2,y=2x 2 1,y= x 2 2,y= x 2 2 (D) y = 2 x 2 7,y= x 2 1,y= x 2 2,y= x 2 2 5

6 The equation y = ax 2 represents a parabola. It is in the form of a U symbol The general formula for a parabola is (y k) 2 =4a(x h). But here it is in the reverse. So we have (1/a)y = x 2 The width of the parabola is determined based on the constant value a So the question is to arrange the values from widest to narrowest. This means that a value is largest at the start and then decreases A simple way to approach this problem is to calculate the ratio of values such as 7/, 2/, 1/2 Since the ratio is (1/a) for the y term, we have all the numbers in the reverse order Soforthefirstoptionitisx 2 = (/7) y, x 2 =(/2)y, x 2 =(2)y,x 2 = (1/2) y Likewise if we calculate for all the other options, we can see that B is the correct answer Question: Which linear equation represents the line passing through points R and S? (A) y = 1.5x 4.5 (B) y = 1.5x (C) y = 0.5x 4.5 (D) y = 0.5x Point R has the coordinates as (5,) Point S has the coordinates as (, 9) Slope (m) has the formula (y 2 y 1 ) / (x 2 x 1 ) We get m = 12/ 8 = /2 Now we plug the value of m in the line equation y y 1 = m(x x 1 ) y = /2 (x 5) y = 1.5 x 4.5 Question: Which line appears to have a slope of zero? 6

7 (A) n (B) k (C) w (D) p Now slope is rise over run or y/x. Rise is the change in y axis and Run is change in x axis So m = 0 means y = 0. This is because if x = 0 then we have m = y / 0 which is an undefined solution So check the line which has no y which otherwise means there is no y term but just x. We can see that w is the line which has no rise Question: Graph the function P (x) = x 4 2 x We need to find the x and y intercept of P (x) To find the x intercept, factorize P (x) It has x 4 term inside it. The easiest way to factor it is to assume x 2 = r Now P(x) becomes r 2 2r + 1 This is equal to (r 1) 2 or P (x) = (x 2 1) 2 But we can reduce x 2 1 further. It becomes (x 1) (x + 1) P (x) = (x 1) 2 (x + 1) 2 Equate P (x) to 0 to find the x intercept We get x = 1,1, 1, 1 because of the square term Now to find the y intercept we substitute x = 0 So now we have y = 1 Therefore now we have x intercept of ( 1,0) and (1,0) and the y intercept of (1,0) This is a quadratic equation and we have a parabola We can find other values by assuming various values of x to get P(x) Question: Graph the solution to y <2x + The equation is already in y = mx + b form Shade the line below the line since y has a less than symbol Question: 7

8 There are n stages as mentioned in the problem So put n = 1 in all the choices. We see that for all the options we get 2 Now put n = 2, we have 6 for (n 2 +n)and(4n 2) Now put n =, we have 12 only for (n 2 +n)which is the correct option Question: There are many ways to solve this but the best solution is to find the ratio of the radius and compare it with the respective heights For option A, we have radius ratio 12/4.25 = 2.82 while the height ratio as 8.5/6 = 1.41 ForoptionB,wehaveradiusratio7.25/4.25= 1.71 while the height ratio as 9/6 = 1.5 For option C, we have radius ratio 6.8/4.25 = 1.6 while the height ratio as 9.6/6 = 1.6 So C is the correct option Question: The table below shows the results of rolling a fair number cube 50 times during a classroom activity. What is the difference between the theoretical probability of rolling a number less than 4 and the experimental results recorded in the table above? (A) 8% (B) 79% (C) 58% (D) 29% The easiest is to calculate the experimental probability It is given that 1, 2, outcome has 7, 12, 10 frequency There are totally 50 trials. So the experimental probability = # outcomes = = 0.58 #oftrials 50 Theoreticalprobability = # favorable outcomes # of possible outcomes The theoretical probability of rolling a number less than 4 will be the sum of probabilities of rolling 1,2 or which are 1/6 each. We add those together [1/6+1/6+1/6] and get /6 or ½ or 0.5 Experimental Theoretical probability = This is equal to Convert it to %, we get 8% 8

9 Question: What is the probability of flipping coins and having 2 of them come up heads and 1 come up tails? (A) 1/ (B) /8 (C) 1/2 (D) 5/8 (E) 7/9 A coin can either have Head(H) or Tail(T). So 2 possible outcomes When 2 coins are flipped it has HH, HT, TH, TT or 4 possible outcomes or 2 2 choices Now for coins the outcome will be 8 or 2 choices Ofthe8choicestherecanbejusttwoHandone T Soitcanbedoneinwayswitheithercoin1and 2asHorcoin2andasHorcoin1andasH So the probability is /8 9