Four prisms

Question No. 1: The prism has the dimensions a = 2.5 cm, b = 100 mm, c = 12 cm. What is its volume?
a) 3000 cm2
b) 300 cm2
c) 3000 cm3
d) 300 cm3

Question No.2: The base of the prism is a rhombus with a side length of 30 cm and a height of 27 cm. The height of the prism is 5dm. What is the volume of the prism?
a) 20 250
b) 450
c) 40,500
d) 4,050

Question No.3: A regular square prism with a square base has a volume of 63 cm3 and a height of 7 cm. Calculate its surface.
a) 756
b) 102
c) 63
d) 414

Question No.4: Calculate the surface of a square prism high vh = 2 dm, the base of which is a trapezoid with bases z1 = 10cm, z2 = 8cm, height v = 4cm and with arms long r1 = r2 = 5cm.
a) 128 cm2
b) 596 cm2
c) 632 cm2
d) 532 cm2

Correct result:

V1 =  300 cm3
V2 =  30000 cm3
S3 =  102 cm2
S4 =  632 cm2

Solution:

a1=2.5 cm b1=100 mm→cm=100/10 cm=10 cm c1=12 cm  V1=a1⋅ b1⋅ c1=2.5⋅ 10⋅ 12=300 cm3a_{1}=2.5 \ \text{cm} \ \\ b_{1}=100 \ mm \rightarrow cm=100 / 10 \ cm=10 \ cm \ \\ c_{1}=12 \ \text{cm} \ \\ \ \\ V_{1}=a_{1} \cdot \ b_{1} \cdot \ c_{1}=2.5 \cdot \ 10 \cdot \ 12=300 \ \text{cm}^3
a2=30 cm b2=20 cm c2=5 dm→cm=5⋅ 10 cm=50 cm  V2=a2⋅ b2⋅ c2=30⋅ 20⋅ 50=30000 cm3a_{2}=30 \ \text{cm} \ \\ b_{2}=20 \ \text{cm} \ \\ c_{2}=5 \ dm \rightarrow cm=5 \cdot \ 10 \ cm=50 \ cm \ \\ \ \\ V_{2}=a_{2} \cdot \ b_{2} \cdot \ c_{2}=30 \cdot \ 20 \cdot \ 50=30000 \ \text{cm}^3
V3=63 cm3 c=7 cm  S1=V3/c=63/7=9 cm2 a=S1=9=3 cm  S3=4⋅ a⋅ c+2⋅ S1=4⋅ 3⋅ 7+2⋅ 9=102 cm2V_{3}=63 \ \text{cm}^3 \ \\ c=7 \ \text{cm} \ \\ \ \\ S_{1}=V_{3}/c=63/7=9 \ \text{cm}^2 \ \\ a=\sqrt{ S_{1} }=\sqrt{ 9 }=3 \ \text{cm} \ \\ \ \\ S_{3}=4 \cdot \ a \cdot \ c + 2 \cdot \ S_{1}=4 \cdot \ 3 \cdot \ 7 + 2 \cdot \ 9=102 \ \text{cm}^2
v4=2 dm→cm=2⋅ 10 cm=20 cm z1=10 cm z2=8 cm v=4 cm r1=5 cm r2=5 cm  S44=z1+z22⋅ v=10+82⋅ 4=36 cm2  S4=(z1+z2+r1+r2)⋅ v4+2⋅ S44=(10+8+5+5)⋅ 20+2⋅ 36=632 cm2v_{4}=2 \ dm \rightarrow cm=2 \cdot \ 10 \ cm=20 \ cm \ \\ z_{1}=10 \ \text{cm} \ \\ z_{2}=8 \ \text{cm} \ \\ v=4 \ \text{cm} \ \\ r_{1}=5 \ \text{cm} \ \\ r_{2}=5 \ \text{cm} \ \\ \ \\ S_{44}=\dfrac{ z_{1}+z_{2} }{ 2 } \cdot \ v=\dfrac{ 10+8 }{ 2 } \cdot \ 4=36 \ \text{cm}^2 \ \\ \ \\ S_{4}=(z_{1}+z_{2}+r_{1}+r_{2}) \cdot \ v_{4} + 2 \cdot \ S_{44}=(10+8+5+5) \cdot \ 20 + 2 \cdot \ 36=632 \ \text{cm}^2

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