Question 1: Recall that in a cuboid with length l, width w, and height h, the face diagonals are calculated using the Pythagoras theorem. Write down the formula for the space diagonal d of the cuboid.

Answer: d = sqrt(l^2 + w^2 + h^2)

Question 2: A cuboid has dimensions 3 cm by 4 cm by 12 cm. Calculate the length of its space diagonal.

Answer: d = sqrt(3^2 + 4^2 + 12^2) = sqrt(9 + 16 + 144) = sqrt(169) = 13 cm

Question 3: Find the face diagonal of a cuboid with length 8 cm and width 6 cm.

Answer: sqrt(8^2 + 6^2) = sqrt(64 + 36) = sqrt(100) = 10 cm

Question 4: A cuboid has dimensions 5 cm, 12 cm, and 13 cm. Calculate all face diagonals.

Answer: Diagonal1 = sqrt(5^2 + 12^2) = 13 cm; Diagonal2 = sqrt(5^2 + 13^2) = sqrt(25+169)=sqrt(194)≈13.93 cm; Diagonal3 = sqrt(12^2 +13^2)= sqrt(144+169)= sqrt(313)≈17.69 cm

Question 1: Calculate the surface area of a cuboid measuring 7 cm by 3 cm by 2 cm.

Answer: Surface area = 2 (lw + lh + wh) = 2 (7*3 + 7*2 + 3*2) = 2 (21 + 14 + 6) = 2 * 41 = 82 cm^2

Question 2: A cuboid has volume 180 cm^3 and length 6 cm. Find its width and height if they are equal.

Answer: Let w=h, then volume = l*w*h=6*w^2=180 => w^2=30 => w=√30≈5.48 cm

Question 3: Determine the volume of a cuboid with dimensions 9 cm, 4 cm, and 3 cm.

Answer: Volume = 9*4*3=108 cm^3

Question 4: Find the total surface area of a cuboid with length 10 cm, width 5 cm, and height 2 cm.

Answer: Surface area = 2(10*5 + 10*2 + 5*2) = 2(50 + 20 + 10)=2*80=160 cm^2

Question 5: Calculate the length of the space diagonal of a cuboid with dimensions 8 cm, 15 cm, and 17 cm.

Answer: d = sqrt(8^2 + 15^2 + 17^2) = sqrt(64 + 225 + 289) = sqrt(578)≈24.04 cm

Question 1: A storage box is a cuboid measuring 2 m by 1.5 m by 0.75 m. What is the length of the space diagonal? Round to two decimal places.

Answer: d = sqrt(2^2 + 1.5^2 + 0.75^2) = sqrt(4 + 2.25 + 0.5625) = sqrt(6.8125) ≈ 2.61 m

Question 2: A rectangular box has a volume of 2400 cm^3. Its length is 20 cm, and width is 15 cm. Find its height and the length of the space diagonal.

Answer: Height = volume / (length*width)=2400 / (20*15)=2400/300=8 cm; Diagonal = sqrt(20^2 + 15^2 + 8^2)= sqrt(400+225+64)= sqrt(689)≈26.27 cm

Question 3: A cuboid has a surface area of 94 cm^2. If its length and width are 4 cm and 3 cm respectively, find its height.

Answer: Surface area = 2(lw + lh + wh) = 94 => 2(12 + 4h + 3h)=94 => 2(12 + 7h)=94 => 24 +14h=47 =>14h=23 => h=23/14≈1.64 cm

Question 1: A shipping crate is a cuboid measuring 1.2 m by 0.8 m by 0.6 m. If the crate must be placed on a shelf with a clearance of 1.5 m, can it fit? Justify your answer using the space diagonal.

Answer: Diagonal = sqrt(1.2^2 + 0.8^2 + 0.6^2)= sqrt(1.44 + 0.64 + 0.36)= sqrt(2.44)≈1.56 m; since 1.56 > 1.5, it cannot fit.

Question 2: A water tank in the shape of a cuboid has dimensions 2 m by 2 m by 3 m. How much water in cubic meters can it hold if it is completely filled?

Answer: Volume = 2*2*3=12 m^3

Question 1: A cuboid has a space diagonal of 25 cm. If its length is 7 cm and width is 24 cm, find its height.

Answer: Using d = sqrt(l^2 + w^2 + h^2): 25 = sqrt(7^2 + 24^2 + h^2) => 625 = 49 + 576 + h^2 => h^2=625 - 625=0, so h=0 (invalid, check calculation). Correct approach: 25^2 = 7^2 + 24^2 + h^2 => 625=49+576+h^2 => h^2=625-625=0, which indicates h=0 is impossible; hence, check for calculation mistake. Correct: 7^2=49, 24^2=576, total 625, so h=0. The provided diagonal cannot be 25 cm with these dimensions; revise question or answer accordingly.

Question 2: Design a cuboid with dimensions that give a surface area of exactly 200 cm^2 and a space diagonal of less than 20 cm. Provide one possible set of dimensions.

Answer: Example: length=8 cm, width=6 cm, height ≈6.7 cm (approximate), satisfying surface area and diagonal constraints (d≈sqrt(8^2+6^2+6.7^2)=≈13.8 cm).

Question 1: A student calculates the face diagonal of a cuboid with length 5 cm and width 12 cm as 13 cm, which is correct. However, they then incorrectly claim that the space diagonal is also 13 cm when dimensions are 5 cm, 12 cm, and 13 cm. Identify and explain the mistake.

Answer: The mistake is assuming the space diagonal is equal to the face diagonal. The space diagonal is calculated using all three dimensions: sqrt(5^2 + 12^2 + 13^2) = sqrt(25 + 144 + 169) = sqrt(338) ≈ 18.38 cm.