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The volume of a cube is 64 cm³. What is its total surface area?
Answer: Let the side of the cube be $a$.
Volume = $a^3 = 64 \Rightarrow a = 4$ cm.
The Total Surface Area (TSA) of a cube is $6a^2$.
TSA = $6(4^2) = 6(16) = 96$ cm².
How many small cubes with side 20 cm can be packed tight in a cubical box with side 2 m?
Answer: First, convert the side of the large box to cm: $2$ m = $200$ cm.
Volume of the large box = $(200)^3 = 8,000,000$ cm³.
Volume of one small cube = $(20)^3 = 8,000$ cm³.
Number of small cubes = $\frac{8,000,000}{8,000} = 1000$ cubes.
The dimensions of a godown are $40 \text{ m} \times 25 \text{ m} \times 10 \text{ m}$. If it is filled with cuboidal boxes, each of dimensions $2 \text{ m} \times 1.25 \text{ m} \times 1 \text{ m}$, then find the number of boxes.
Answer: Volume of the godown = $40 \times 25 \times 10 = 10,000$ m³.
Volume of one box = $2 \times 1.25 \times 1 = 2.5$ m³.
Number of boxes = $\frac{10,000}{2.5} = 4000$ boxes.
Two cubes each of volume 125 cm³ are joined end to end. Find the surface area of the resulting cuboid.
Answer: Volume of one cube = $a^3 = 125 \Rightarrow a = 5$ cm.
When joined end to end, the resulting cuboid has:
Length ($l$) = $5 + 5 = 10$ cm
Width ($w$) = $5$ cm
Height ($h$) = $5$ cm
Total Surface Area = $2(lw + wh + hl) = 2((10)(5) + (5)(5) + (5)(10)) = 2(50 + 25 + 50) = 2(125) = 250$ cm².
A cube of side 4 cm is cut into cubes of side 1 cm. What is the ratio of the surface areas of the original cube and all the cut-out cubes?
Answer: Surface area of the original cube = $6(4^2) = 6(16) = 96$ cm².
Volume of the original cube = $4^3 = 64$ cm³, which yields 64 smaller cubes of 1 cm³ each.
Surface area of one 1 cm cube = $6(1^2) = 6$ cm².
Total surface area of all 64 smaller cubes = $64 \times 6 = 384$ cm².
Ratio = $96 : 384 = 1 : 4$.
The surface areas of the three faces of a cuboid that meet at one of the corners of the cuboid are 6 cm², 15 cm², and 10 cm² respectively. What is the volume of the cuboid?
Answer: Let the dimensions be $l, w, h$.
Given: $lw = 6$, $wh = 15$, $lh = 10$.
Multiply all three equations: $(lw)(wh)(lh) = 6 \times 15 \times 10$
$(lwh)^2 = 900$
Volume $V = lwh = \sqrt{900} = 30$ cm³.
A cube of side 5 cm is painted on all its faces. If it is sliced into 1 cm³ cubes, how many of these 1 cm³ cubes have:
(i) exactly three faces painted?
(ii) exactly two faces painted?
(iii) exactly one face painted?
(iv) no face painted?
Answer: Total smaller cubes = $5 \times 5 \times 5 = 125$.
(i) Exactly three faces painted (corners): 8 cubes.
(ii) Exactly two faces painted (edges, excluding corners): 12 edges $\times$ (5 - 2) = $12 \times 3 = $ 36 cubes.
(iii) Exactly one face painted (centers of faces): 6 faces $\times$ (3 $\times$ 3) = $6 \times 9 = $ 54 cubes.
(iv) No face painted (interior cubes): $(5 - 2) \times (5 - 2) \times (5 - 2) = 3 \times 3 \times 3 = $ 27 cubes.
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