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A prism is a solid figure formed from two congruent polygons with their corresponding sides parallel (the bases) and the parallelogram (lateral faces) formed by joining the corresponding vertices of the polygons. The lines joining the vertices of the polygons are lateral edges. Prisms are named according to the base - for example, a triangular prism has two triangular bases (and three lateral faces); a quadrangular prism has bases that are quadrilaterals. Pen­tagonal, hexagonal, etc. prism have bases that are pentagons, hexagons, etc.

A right prism is one in which the lateral edges are at right angles to the bases (i.e. the lateral faces are rectangles) - otherwise the prism is an oblique prism (i.e. one base is displaced with respect to the other, but remains parallel to it). If the bases are regular polygons and the prism is also a right prism, then it is a regular prism.

A cone is a solid figure formed by a closed plane curve on a plane (the base) and all the lines joining points of the base to a fixed point (the vertex) not in the plane of the base. The closed curve is the directrix of the cone and the lines to the vertex are its generators (or elements). The curved area of the cone forms its lateral surface. Cones are named according to the base, e.g. a circular cone or an elliptical cone. If the base has a center of symmetry, a line from the vertex to the center is the axis of the cone. A cone that has its axis perpendicular to its base is a right cone; otherwise the cone is a oblique cone. The altitude of a cone (h) is the perpendicular distance from the plane of the base to the vertex. The volume of any cone is l/3hA, where A is the area of the base. A right circular cone (circular base with perpendicular axis) has a slant height (s), equal to the distance from the edge of the base to the vertex (the length of a generator). The term "cone" is often used loosely for "conical sur­face".

A pyramid is a solid figure (a polyhedron) formed by a polygon (the base) and a number of triangles (lateral faces) with a common vertex that is not coplanar with the base. Line segments from the common vertex to the vertices of the base are lateral edges of the pyramid. Pyramids are named according to the base: a triangular pyramid (which is a tetrahedron), a square pyramid, a pentagonal pyramid, etc.

If the base has a center, a line from the center to the vertex is the axis of the pyramid. A pyramid that has its axis perpendicular to its base is a right pyramid; otherwise, it is an oblique pyramid, then it is also a regular pyramid.

The altitude (h) of a pyramid is the perpendicular distance from the base to the vertex. The volume of any pyramid is l/3Ah, where A is the area of the base. In a regular pyramid all the lateral edges have the same length. The slant height (s) of the pyramid is the altitude of a face; the total surface area of the lateral faces is l/2sp, where p is the perimeter of the base polygon.
XII. POLYHEDRON
A polyhedron is a surface composed of plane polygonal surfaces (faces). The sides of the polygons, joining two faces, are its edges. The corners, where three or more faces meet, are its vertices. Generally, the term

"polyhedron" is used for closed solid figure. A convex polyhedron is one for which a plane containing any face does not cut other faces; otherwise the polyhedron is concave.

A regular polyhedron is one that has identical (congruent) regular polygons forming its faces and has all its polyhedral angles congruent. There are only five possible convex regular polyhedra:

1. 
   tetrahedron - four triangular faces,
   
2. 
   cube - six square faces,
   
3. 
   octahedron - eight triangular faces,
   
4. 
   dodecahedron - twelve pentagonal faces,
   
5. 
   icosahedron - twenty triangular faces.
   

A uniform polyhedron is a polyhedron that has identical polyhedral angles at all its vertices, and has all its faces formed by regular polygons (not necessarily of the same type). The five regular polyhedra are also uniform polyhedra. Right prisms and antiprisms that have regular polygons as bases are also uniform. In addition, there are thirteen semiregular polyhedra, the so-called Archimedian solids. For example, the icosidodecahedron has 32 faces - 20 triangles and 12 pentagons. It has 60 edges and 30 vertices, each vertex beeing the meeting point of two triangles and two pentagons. Another example is the truncated cube, obtained by cutting the corners off a cube. If the corners are cut so that the new vertices lie at the centers of the edges of the original cube, a cuboctahedron results. Truncating the cuboctahedron and "distorting" the rectangular faces into squares yields another Archimedian solid. Other uniform polyhedra can be generated by truncating the four other regular polyhedra or the icosidodecahedron.

9. We should like to show that the Pythagorean Property is true for all right triangles, there being several proofs of this property. 10. Let us discuss one of them. 11. Before giving the proof let us state the Pythagorean Property in mathematical language. 12. In the triangle above, c represents the measure of the hypotenuse, and a and b represent the measures of the other two sides.

13. If we con­struct squares on the three sides of the triangle, the area-measure will be a², b² and c². 14. Then the Pythagorean Property could be stated as follows: c² = a² + b². 15. This proof will involve working with areas. 16. To prove that c² = a² + b² for the triangle above, construct two squares each side of which has a measure a + b as shown above. 17. Separate the first of the two squares into two squares and two rectangles as shown. 18. Its total area is the sum of the areas of the two squares and the two rectangles.

19. In the second of the two squares construct four right trian­gles. 20. Are they congruent? 21. Each of the four triangles being congruent to the original triangle, the hypotenuse has a measure c. 22. It can be shown that PQRS is a square, and its area is c². 23. The total area of the second square is the sum of the areas of the four triangles and the square PQRS. A = c²+4 (½ ab). The two squares being congruent to begin with², their area measures are the same. 25. Hence we may conclude the following:

26. By subtracting 2ab from both area measures we obtain a²+ b² = c² which proves the Pythagorean Property for all right trian­gles.

6. You will know the number represented by c if you can find a num­ber which, when used as a factor twice, gives a product of 25. 7. Of course, 5x5 = 25, so c = 5 and 5 is called the positive square root (корень) of 25. 8. If a number is a product of two equal factors, then either (любой) of the equal factors is called a square root of the number. 9. When we say that y is the square root of K we merely (вcero лишь) mean that y² = K. 10. For example, 2 is a square root of 4 because 2² = 4. 11. The product of two negative numbers being a positive number, —2 is_also a square root of 4 because (—2)² = 4. The following symbol √ called a radical sign is used to denote the positive square root of a number. 13. That is √K means the positive square root of K. 14. Therefore √4 =2 and √25 = 5. 15. But suppose you wish to find the √20. 16. There is no integer whose square is 20, which is obvious from the following computation. 4²= 16 so √16 = 4; a² = 20 so 4<a<5, 5² = 25, so √25 = 5. 17. √20 is greater than 4 but less than 5. 18. You might try to get a closer approxi­mation of √20 by squaring some numbers between 4 and 5. 19. Since √20 is about as near to 4² as¹ to 5², suppose we square 4.4 and 4.5.

4.4²= 19.36 a² = 20 4.5² = 20.25

20. Since 19.36<20<20.25 we know that 4.4<a<4_:5. 21. 20 being nearer to 20.25 than to 19.36, we might guess that √20 is nearer to 4.5 than to 4.4. 22. Of course, in order to make sure² that √20 = 4.5, to the nearest tenth, you might select values between 4.4 and 4.5, squ­are them, and check the results. 23. You could continue the process indefinitely and never get the exact value of 20. 24. As a matter of fact, √20 represents an irrational number which can only be expressed approximately as rational number. 25. Therefore we say that √20 = 4.5 approximately (to the nearest tenth).

1. 
   A trip takes 6 hours to complete. After traveling ¼ of an hour, 1⅜ hours, and 2⅓ hours, how much time does one need to complete the trip?
   

1. 
   2 ¹/¹² hours
   
2. 
   2 hours, 2½ minutes
   
3. 
   2 hours, 5 minutes
   
4. 
   2⅛ hours
   
5. 
   2 hours, 7½ minutes
   

2. 
   It takes 30 days to fill laboratory dish with bacteria. If the size of the bacteria doubles each day, how long did it take for the bacteria to fill one half of the dish?
   

1. 
   10 days
   
2. 
   15 days
   
3. 
   24 days
   
4. 
   29 days
   
5. 
   29.5 days
   

3. 
   A car wash can wash 8 cars in 18 minutes. At this rate how many cars can the car wash wash in 3 hours?
   

1. 
   13
   
2. 
   40.5
   
3. 
   80
   
4. 
   125
   
5. 
   405
   

4. 
   If the ratio of the areas of 2 squares is 2 : 1, then the ratio of the perimeters of the squares is
   

1. 
   1: 2
   
2. 
   1: √2
   
3. 
   √2 : 1
   
4. 
   2 : 1
   
5. 
   4: 1
   

5. There are three types of tickets available for a concert: orchestra, which cost $12 each; balcony, which cost $9 each; and box, which cost $25 each. There were P orchestra tickets, B balcony tickets, and R box tickets sold for the concert. Which of the following expressions gives the percentage of the ticket proceeds due to the sale of orchestra tickets?

(A) 100 x 

6. City B is 5 miles east of City A. City C is 10 miles southeast of City B. Which of the following is the closest to the distance from City A to City C?

1. 
   11 miles
   
2. 
   12 miles
   
3. 
   13 miles
   
4. 
   14 miles
   
5. 
   15 miles
   

7. 
   If 3x – 2y = 8, then 4y – 6x is:
   

1. 
   -16
   
2. 
   -8
   
3. 
   8
   
4. 
   16
   
5. 
   cannot be determined
   

8. 
   It costs 10c. a kilometer to fly and 12c. a kilometer to drive. If you travel 200 kilometers, flying x kilometers of the distance and driving the rest, then the cost of the trip in dollars is:
   

1. 
   20
   
2. 
   24
   
3. 
   24 – 2x
   
4. 
   24 – 0.02x
   
5. 
   2.400 – 2x
   

9. 
   If the area of a square increases by 69%, then the side of the square increases by:
   

1. 
   13%
   
2. 
   30%
   
3. 
   39%
   
4. 
   69%
   
5. 
   130%
   

10. 
    There are 30 socks in a drawer. 60% of the socks are red and rest are blue. What is the minimum number of socks that must be taken from the drawer without looking in order to be certain that at least two blue socks have been chosen?
    

1. 
   2
   
2. 
   3
   
3. 
   14
   
4. 
   16
   
5. 
   20
   

11. 
    How many squares with sides ½ inch long are needed to cover a rectangle that is 4 feet long and 6 feet wide?
    

1. 
   24
   
2. 
   96
   
3. 
   3,456
   
4. 
   13,824
   
5. 
   14,266
   

12. 
    In a group of people solicited by a charity, 30% contributed $40 each, 45 % contributed $20 each, and the rest contributed $12 each. What percentage of the total contributed came from people who gave $40?
    

1. 
   25%
   
2. 
   30%
   
3. 
   40%
   
4. 
   45%
   
5. 
   50%
   

13. 
    A trapezoid ABCD is formed by adding the isosceles right triangle BCE with base 5 inches to the rectangle ABED where DE is t inches. What is the area of the trapezoid in square inches?
    

1. 
   5t + 12.5
   
2. 
   5t + 25
   
3. 
   2.5t + 12.5
   
4. 
   (t + 5)²
   
5. 
   t² + 25
   

14. 
    A manufacturer of jam wants to make a profit of $75 by selling 300 jars of jam. It costs 65c. each to make the first 100 jars of jam and 55c. each to make each jar after the first 100. What price should be charged for the 300 jars of jam?
    

1. 
   $75
   
2. 
   $175
   
3. 
   $225
   
4. 
   $240
   
5. 
   $250
   

15. 
    A car traveled 75% of the way from town A to town B by traveling for T hours at an average speed of V mph. The car travels at an average speed of S mph for the remaining part of the trip. Which of the following expressions represents the time the car traveled at S mph?
    

1. 
   VT/ S
   
2. 
   VS/4T
   
3. 
   4VT/3S
   
4. 
   3S/VT
   
5. 
   VT/3S
   

16. 
    A company makes a profit of 7% selling goods which cost $2,000; it also makes a profit of 6% selling a machine that cost the company $5,000. How much total profit did the company make on both transactions?
    

1. 
   $300
   
2. 
   $400
   
3. 
   $420
   
4. 
   $440
   
5. 
   $490
   

17. 
    The ratio of chickens to pigs to horses on a farm can be expressed as the triple ratio 20: 4: 6. If there are 120 chickens on the farm, then the number of horses on the farm is
    

1. 
   4
   
2. 
   6
   
3. 
   24
   
4. 
   36
   
5. 
   60
   

18. 
    If x² - y² = 15 and x + y =3, then x – y is
    

1. 
   – 3
   
2. 
   0
   
3. 
   3
   
4. 
   5
   
5. 
   cannot be determined
   

19. What is the area of the shaded region? The radius of the outer is a and the radius of each of the circles inside the large circle is a/3.

1. 
   0
   
2. 
   (⅓)a²
   
3. 
   (⅔)a²
   
4. 
   (⁷/₉)a²
   
5. 
   (⁸/₉)a²
   

1. 
   4
   
2. 
   6
   
3. 
   8
   
4. 
   10
   
5. 
   12
   

21. A warehouse has 20 packers. Each packer can load ⅛ of a box in 9 minutes. How many boxes can be loaded in 1½ hours by all 20 packers?

1. 
   1¼
   
2. 
   10¼
   
3. 
   12½
   
4. 
   20
   
5. 
   25
   

22. In Motor City 90% of the population own a car, 15 % own a motorcycle, and everybody owns one or the other or both. What is the percentage of motorcycle owners who own cars?

1. 
   5 %
   
2. 
   15 %
   
3. 
   33⅓ %
   
4. 
   50 %
   
5. 
   90 %
   

23. Towns A and C are connected by a straight highway which is 60 miles long.

The straight-line distance between town A and town B is 50 miles, and the straight-line distance from town B to town C is 50 miles. How many miles is it from town B to the point on the highway connecting towns A and C which is closest to town B?

1. 
   30
   
2. 
   40
   
3. 
   30√2
   
4. 
   50
   
5. 
   60
   

24. A chair originally cost $ 50.00. The chair was offered for sale at 108% of its cost. After a week the price was discounted 10% and the chair was sold. The chair was sold for

1. 
   $45.00
   
2. 
   $48.60
   
3. 
   $49.00
   
4. 
   $49.50
   
5. 
   $54.00
   

25. A worker is paid x dollars for the first 8 hours he works each day. He is paid y dollars per hour for each hour he works in excess of 8 hours. During one week he works 8 hours on Monday, 11 hours on Tuesday, 9 hours on Wednesday, 10 hours on Thursday, and 9 hours on Friday. What is his average daily wage in dollars for the five-day week?

(A) x + (7/5) y

(B) 2x + y

(C) (5x + 8y)/ 5

(D) 8x + (7/5) y

1. 
   112,896
   
2. 
   3,136
   
3. 
   720
   
4. 
   112
   
5. 
   9
   

27. 
    A motorcycle costs $ 2,500 when it is brand new. At the end of each year it is worth ⁴/₅ of what it was at the beginning of the year. What is the motorcycle worth when it is three years old?
    

1. 
   $1,000
   
2. 
   $1,200
   
3. 
   $1,280
   
4. 
   $1,340
   
5. 
   $1,430
   

27. 
    If x + 2y = 2x + y, then x – y is equal to
    

1. 
   0
   
2. 
   2
   
3. 
   4
   
4. 
   5
   
5. 
   cannot be determined
   

27. 
    Mary, John, and Karen ate lunch together. Karen’s meal cost 50% more than John’s meal and Mary’s meal cost ⁵/₆ as much as Karen’s meal. If Mary paid $2 more than John, how much was the total that the three of them paid?
    

1. 
   $2833
   
2. 
   $30.00
   
3. 
   $35.00
   

1. 
   $37.50
   
2. 
   $40.00
   

27. 
    If the angles of triangle are in the ratio 1 : 2 :2, then the triangle
    

1. 
   is isosceles
   
2. 
   is obtuse
   
3. 
   is a right triangle
   
4. 
   is equilateral
   
5. 
   has one angle greater than 80º
   

27. 
    Successive discounts of 20% and 15% are equal to a single discount of
    
    1. 
       30%
       
    2. 
       32%
       
    3. 
       34%
       
    4. 
       35%
       
    5. 
       36%
       

27. 
    It takes Eric 20 minutes to inspect a car. Jane only needs 18 minutes to inspect a car. If they both start inspecting cars at 8:00 a.m., what is the first time they will finish inspecting a car at the same time?
    
    1. 
       9:30 a.m.
       
    2. 
       9:42 a.m.
       
    3. 
       10:00 a.m.
       
    4. 
       11:00 a.m.
       
    5. 
       2:00 p.m.
       

27. 
    If x/y = 4 and y is not 0, what percentage (to the nearest percent) of x is
    

1. 
   25
   
2. 
   57
   
3. 
   75
   
4. 
   175
   
5. 
   200
   

27. 
    If x > 2 and y > - 1, then
    
    1. 
       xy > - 2
       
    2. 
       – x < 2y
       
    3. 
       xy < - 2
       
    4. 
       –x > 2y
       
    5. 
       x < 2y
       

27. 
    If x = y =2z and x●y● z = 256, then x equals
    
    1. 
       2
       
    2. 
       2 ³√2
       
    3. 
       4
       
    4. 
       4 ³√2
       
    5. 
       8
       

27. 
    If the side of square increases by 40%, then the area of the square increases by
    
    1. 
       16%
       
    2. 
       40%
       
    3. 
       96%
       
    4. 
       116%
       
    5. 
       140%
       

27. 
    If 28 cans of soda cost $21.00, then 7 cans of soda should cost
    
    1. 
       $5.25
       
    2. 
       $5.50
       
    3. 
       $6.40
       
    4. 
       $7.00
       
    5. 
       $10.50
       

27. 
    If the product of 3 consecutive integers is 210, then the sum of the two smaller integers is
    
    1. 
       5
       
    2. 
       11
       
    3. 
       12
       
    4. 
       13
       
    5. 
       18
       

27. 
    Both circles have radius 4 and the area enclosed by both circles is 28. What is the area of the shaded region?
    

1. 
   0
   
2. 
   2
   
3. 
   4
   
4. 
   4²
   
5. 
   16
   

27. 
    If a job takes 12 workers 4 hours to complete, how long should it take 15 workers to complete the job?
    
    1. 
       2 hr 40 min
       
    2. 
       3 hr
       
    3. 
       3 hr 12 min
       
    4. 
       3 hr 24 min
       
    5. 
       3 hr 30 min
       

27. 
    If a rectangle has length L and the width is one half of the length, then the area of the rectangle is
    
    1. 
       L
       
    2. 
       L²
       
    3. 
       ½ L²
       
    4. 
       ¼ L²
       
    5. 
       2L
       

27. 
    What is the next number in the arithmetic progression 2, 5, 8 .......?
    
    1. 
       7
       
    2. 
       9
       
    3. 
       10
       
    4. 
       11
       
    5. 
       12
       

27. 
    The sum of the three digits a, b, and c is 12. What is the largest three-digit number that can be formed using each of the digits exactly once?
    
    1. 
       921
       
    2. 
       930
       
    3. 
       999
       
    4. 
       1,092
       
    5. 
       1,200
       

27. 
    What is the farthest distance between two points on a cylinder of height 8 and the radius 8?
    
    1. 
       8 √2
       
    2. 
       8 √3
       
    3. 
       16
       
    4. 
       8 √5
       
    5. 
       8 (2 + 1)
       

27. 
    For which values of x is x² - 5x + 6 negative?
    
    1. 
       x < 0
       
    2. 
       0 < x < 2
       
    3. 
       2 < x < 3
       
    4. 
       3 < x < 6
       
    5. 
       x > 6
       

27. 
    A plane flying north at 500 mph passes over a city at 12 noon. A plane flying east at the same time altitude passes over the same city at 12:30p.m. The plane is flying east at 400 mph. To the nearest hundred miles, how far apart are the two planes at 2 p.m.?
    
    1. 
       600 miles
       
    2. 
       1,000 miles
       
    3. 
       1,100 miles
       
    4. 
       1,200 miles
       
    5. 
       1,300 miles
       

27. 
    A manufacturer of boxes wants to make a profit of x dollars. When she sells 5,000 boxes it costs 5¢ a box to make the first 1,000 boxes and then it costs y¢ a box to make the remaining 4,000 boxes. What price in dollars should she charge for the 5,000 boxes?
    
    1. 
       5,000 + 1,000y
       
    2. 
       5,000 + 1,000y + 100x
       
    3. 
       50 + 10y + x
       
    4. 
       5,000 + 4,000y + x
       
    5. 
       50 +40y + x
       

27. 
    An angle of x degrees has the property that its complement is equal to ¹/₆ of its supplement where x is
    
    1. 
       30
       
    2. 
       45
       
    3. 
       60
       
    4. 
       63
       
    5. 
       72
       

27. 
    The angles of a triangle are in the ratio 2 : 3 : 4. The largest angle in the triangle is
    
    1. 
       30º
       
    2. 
       40º
       
    3. 
       70º
       
    4. 
       75º
       
    5. 
       80º
       

27. 
    If x < y, y < z, and z > w, which of the following statements is always true?
    

(B) x < z

(C) y = w

(D) y > w

(E) x < w

51 ABCD has area equal to 28. BC is parallel to AD. BA is perpendicular to AD. If BC is 6 and AD is 8, then what is CD?

1. 
   2√2
   
2. 
   2√3
   
3. 
   4
   
4. 
   2√5
   
5. 
   6
   

52. Write formulas according to descriptions:

1. 
   a plus b over a minus b is equal to c plus d over c minus d.
   
2. 
   a cubed is equal to the logarithm of d to the base c.
   
3. 
   a)  of z is equal to b, square brackets, parenthesis, z divided by с sub m plus 2. close parenthesis, to the power m over m minus 1, minus 1, close square brackets;
   

4. the absolute value of the quantity  sub j of t one, minus  sub j of t two, is less than or equal to the absolute value of the quantity M of t₁ minus  over j, minus M of t₂ minus  over j.

5. R is equal to the maximum over j of the sum from i equals one to i equals n of the modulus of aij of t, where t lies in the closed interval a b and where j runs from one to n.

6. the limit as n becomes infinite of the integral of f of s and  n of s plus delta n of s, with respect to s, from  to t, is equal to the integral of f of s and  of s, with respect to s, from  to t.

7.  sub n minus r sub s plus 1 of t is equal to p sub n minus r sub s plus 1, times e to the power t times  sub q plus s.

8. 
   L sub n adjoint of g is equal to minus 1 to the n, times the n-th derivative of a sub zero conjugate times g, plus, minus one to the n minus 1, times the n minus first derivative of a sub one conjugate times g, plus ... plus a sub n conjugate times g.
   
9. 
   the partial derivative of F of lambda sub i of t and t, with respect to lambda, multiplied by lambda sub i prime of t, plus the partial derivative of F with arguments lambda sub i of t and t, with respect to t, is equal to 0.
   
10. 
    the second derivative of y with respect to s. plus y, times the quantity 1 plus b of s, is equal to zero.
    
11. 
    f of z is equal to  sub mk hat, plus big 0 of one over the absolute value of z, as absolute z becomes infinite, with the argument of z equal to gamma.
    
12. 
    D sub n minus 1 prime of x is equal to the product from s equal to zero to n of, paranthesis, 1 minus x sub s squared, close paranthesis, to the power epsilon minus 1.
    
13. 
    K of t and x is equal to one over twoi, times the integral of K of t and z, over  minus  of x, with respect to  along curve of the modulus of  minus one half, is equal to rho.
    
14. 
    The second partial (derivative) of u with respect to t, plus a to the fourth power, times the Laplacian of the Laplacian of u, is equal to zero, where a is positive.
    
15. 
    D sub k of x is equal to one over two i, times integral from c minus i infinity to c plus i infinity of dzeta to the k of, , x to the  divided by , with respect to , where c is greater than 1.