Figure for problem 1
1.Answer : A
Explanation:
 Check angle condition: equilateral triangle 60, regular decagon 144, regular
15-gon 156.
 Indeed, 60+144+156=360, so a these polygons fit around a point
 Place polygons 1, 2 and 3 around point a as shown in the figure above
 Then we are obligated to place triangle 4 as shown to cover everything
around B
 Next, we have to place the decagon 4 at C to cover everything around C
 But now we are running into trouble as there is a small angle of 12 degrees
at D which cannot be covered without overlap
 Hence, we get stuck after using only 5 tiles; such a tessellation does not exist.
2. Answer: C
Explanation:
 Area of one lateral face= slant height x base /2 = 611 ft x 751 ft/2
 Lateral surface area = 4x area of one lateral face = 2x611x751=917,722
≈917,700 ft2
3. Answer: B
Explanation:
 Volume of rectangular solid = 15 x15x 9=2025 ft3
 Volume of rectangular pyramid = 1/3 x 15 x15 x8 = 600 ft3
 Total volume = 2025+600 = 2625 ft3
4. Answer: D
Explanation:
 Diameter of the base = 10 in → radius of the base R= 5 in
 Volume =πR2h→ h = Volume/(πR2) =549.5/(3.14x25)=7 in
5. Answer: A
Explanation:
 Denote by L the side length of the square base, by D the diameter of the
cylinder
 Also denote by h the common height of the two solids
 Denote by A= the area of the base of the rectangular solid and by B, the
area of the base of the cylinder. We have that A=L2 and B=πR2
 Since the two volumes are equal we deduce that A x h = B x h, which means
A=B.
 From this we have L2=πR2 from which L=√π R =√π D/2 ≈0.886D from which
L<D
 From the highlighted relations it follows that (i) and (iv) are true, while (ii)
and (iii) are false.
6. Answer: C
Explanation:
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


Denote the radius R=d/2; it follows that h=d =2R.
Volume cone = 1/3 x πR2h=1/3 x πR2 x (2R) =(2π/3)R3
Volume hemisphere = (1/2) x (4π/3)R3=(2π/3)R3
Hence the two volumes are equal; the desired ratio is 1:1
7. Answer: D
Explanation:
 Volume of rectangular solid = 13 x12x 8=1248 ft3
 Volume of rectangular pyramid = 1/3 x 13 x12 x5 = 260 ft3
 Total volume of the tent= 11248+260 = 1508 ft3
8. Answer: B
Explanation:
 The base is an isosceles triangle with sides 8, 8 and base 4.
 The altitude a corresponding to the base in this triangle can be obtained
from the Pythagorean theorem: a2=82-22=60 from which a=2√15
 The area of the base is the ½ x a x 4 = 4√15
 The volume of the prism = Area base x height =8√15≈30.98 ft3
9. Answer: D
Explanation:
 The amount of wrapping paper is proportional to the surface area of each
box.
 The two boxes are similar and the scale ratio is 48/12=4
 Surface area of Larger box/ Surface area of smaller box = (scale ratio)2=16
 Hence the desired ratio is 16:1
10. Answer: None of the options is correct
Explanation:
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



There are two types of points in this arrangement
Type I: 3, 12, 12 point (triangle, dodecagon, dodecagon)
Type II: 3, 4, 3, 12 point (triangle, square, triangle dodecagon)
In conclusion the vertex arrangements are 3, 12, 12 and 3, 4, 3, 12
Although options B and C come close to this, none of them is actually
correct.
11. Answer: B
Explanation:
 There are only 8 semi-regular tessellations – see the figure below.
 In each of these tessellations we have either 2 or 3 types of regular
polygons.
 More precisely, tessellations 1, 2, 3, 5, 6 and 8 have 2 types of regular
polygons.
 On the other hand, tessellations 4 and 7 have 3 types of regular polygons.
3.3.3.3.6
3.3.3.4.4
3.3.4.3.4
3.4.6.4
3.6.3.6
3.12.12
4.6.12
4.8.8
12. Answer : A
Explanation:
 An octahedron has V=6 vertices and E=12 edges. Thus, E=2V so the
statement is true.
13. Answer: A
Explanation:
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


We are given that V=15, E=31
According to Euler’s formula V-E+F=2
It follows that 15-31+F=2 → F= -15+31+2=18
Hence, the polyhedron has 18 faces.
14. Answer: B
Explanation:
 In the formula for the volume of a pyramid, the height h is the distance
from the vertex of the pyramid to the base. The distance measured along
one face of the pyramid to the base is usually called slant height and it is
always larger than h. Thus, height and slant height are two different things.
The statement is false.
15. Answer: B
Explanation:
 Fold first the squares which form a string of four consecutive squares (the
ones in the middle row)
 We obtain the four lateral faces of a cube without the top and the bottom
in place
 Folding down the top square and folding up the bottom square we are
closing the cube.
 In conclusion, the figure can be folded into a cube. The statement is true.