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 prism base is a rhombus with a side length of 30 cm and a height of 27 cm. The height of the prism is 5 dm. 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 answer:

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

Step-by-step explanation:

$$\begin{gathered} a_1=2.5\text{cm} \\ {b}_1=100\text{mm}\to \text{cm}=100:10\text{cm}=10\text{cm} \\ {c}_1=12\text{cm} \\ {V}_1=a_1\cdot b_1\cdot c_1=2.5\cdot 10\cdot 12=300{\text{cm}}^3 \end{gathered}$$

$$\begin{gathered} a_2=30\text{cm} \\ {b}_2=20\text{cm} \\ {c}_2=5\text{dm}\to \text{cm}=5\cdot 10\text{cm}=50\text{cm} \\ {V}_2=a_2\cdot b_2\cdot c_2=30\cdot 20\cdot 50=30000{\text{cm}}^3 \end{gathered}$$

$$\begin{gathered} V_3=63{\text{cm}}^3 \\ c=7\text{cm} \\ {S}_1=V_3/c=63/7=9{\text{cm}}^2 \\ a=\sqrt{S_1}=\sqrt{9}=3\text{cm} \\ {S}_3=4\cdot a\cdot c+2\cdot S_1=4\cdot 3\cdot 7+2\cdot 9=102{\text{cm}}^2 \end{gathered}$$

$$\begin{gathered} v_4=2\text{dm}\to \text{cm}=2\cdot 10\text{cm}=20\text{cm} \\ {z}_1=10\text{cm} \\ {z}_2=8\text{cm} \\ v=4\text{cm} \\ {r}_1=5\text{cm} \\ {r}_2=5\text{cm} \\ {S}_{44}=\frac{z_1+z_2}{2}\cdot v=\frac{10+8}{2}\cdot 4=36{\text{cm}}^2 \\ {S}_4=(z_1+z_2+r_1+r_2)\cdot v_4+2\cdot S_{44}=(10+8+5+5)\cdot 20+2\cdot 36=632{\text{cm}}^2 \end{gathered}$$

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