Step 1: Identify the formula for the volume of a cube, which is $$V = a^3$$, where \(a\) is the side length.
Step 2: Substitute \(a = 5\) cm into the formula: $$V = 5^3 = 5 \times 5 \times 5 = 125$$ cm³.
Step 3: Identify the formula for the total surface area of a cube, which is $$TSA = 6a^2$$.
Step 4: Substitute \(a = 5\) cm into the formula: $$TSA = 6 \times 5^2 = 6 \times 25 = 150$$ cm².
Step 5: Thus, the volume is 125 cm³ and the total surface area is 150 cm².

Step 1: Identify the dimensions of the cuboid: length \(l = 10\) cm, width \(b = 8\) cm, and height \(h = 5\) cm.
Step 2: Use the formula for the Total Surface Area of a cuboid: $$TSA = 2(lb + bh + hl)$$.
Step 3: Substitute the values: $$TSA = 2(10 \times 8 + 8 \times 5 + 5 \times 10)$$.
Step 4: Calculate the products inside the parentheses: $$TSA = 2(80 + 40 + 50)$$.
Step 5: Sum the products: $$TSA = 2(170) = 340$$ cm².

Step 1: Identify the formula for the volume of a cylinder: $$V = \pi r^2 h$$.
Step 2: Use the value of \(\pi \approx \frac{22}{7}\), radius \(r = 7\) cm, and height \(h = 10\) cm.
Step 3: Substitute the values into the formula: $$V = \left(\frac{22}{7}\right) \times 7^2 \times 10$$.
Step 4: Simplify the expression: $$V = 22 \times 7 \times 10$$.
Step 5: Calculate the final volume: $$V = 154 \times 10 = 1,540$$ cm³.

Step 1: Identify the formula for the Curved Surface Area of a cone: $$CSA = \pi rl$$, where \(r\) is the radius and \(l\) is the slant height.
Step 2: Substitute the given values \(r = 3\) cm and \(l = 5\) cm into the formula.
Step 3: Using \(\pi \approx 3.14\): $$CSA = 3.14 \times 3 \times 5$$.
Step 4: Calculate the product: $$CSA = 3.14 \times 15 = 47.1$$ cm².

Step 1: Use the formula for the surface area of a sphere: $$SA = 4\pi r^2$$.
Step 2: Substitute \(r = 21\) cm and \(\pi = \frac{22}{7}\) into the formula.
Step 3: $$SA = 4 \times \left(\frac{22}{7}\right) \times 21 \times 21$$.
Step 4: Simplify the calculation: $$SA = 4 \times 22 \times 3 \times 21$$.
Step 5: Perform the multiplication: $$SA = 88 \times 63 = 5,544$$ cm².

Step 1: The volume of a sphere is proportional to the cube of its radius: $$V \propto r^3$$.
Step 2: Let the old radius be \(r\) and the new radius be \(2r\).
Step 3: The old volume is $$V_{old} = \frac{4}{3}\pi r^3$$.
Step 4: The new volume is $$V_{new} = \frac{4}{3}\pi (2r)^3 = \frac{4}{3}\pi (8r^3)$$.
Step 5: The ratio of new volume to old volume is $$8r^3 : r^3 = 8 : 1$$.

Step 1: Calculate the volume of each of the three cubes: $$V_1 = 3^3 = 27$$, $$V_2 = 4^3 = 64$$, and $$V_3 = 5^3 = 125$$.
Step 2: Find the total volume of the new cube by summing the volumes: $$V_{total} = 27 + 64 + 125 = 216$$ cm³.
Step 3: Let the side of the new cube be \(A\). The volume is $$A^3 = 216$$.
Step 4: Calculate the cube root of 216: $$A = \sqrt[3]{216} = 6$$ cm.

Step 1: Convert all dimensions to centimeters. Tank radius \(R = 1400\) cm, Tank height \(H = 500\) cm. Bowl radius \(r = 7\) cm.
Step 2: Calculate the volume of the cylindrical tank: $$V_{tank} = \pi R^2 H = \pi \times 1400^2 \times 500$$.
Step 3: Calculate the volume of one hemispherical bowl: $$V_{bowl} = \frac{2}{3}\pi r^3 = \frac{2}{3}\pi \times 7^3$$.
Step 4: The number of bowls is $$V_{tank} / V_{bowl} = \frac{\pi \times 1400 \times 1400 \times 500}{\frac{2}{3}\pi \times 7 \times 7 \times 7}$$.
Step 5: Simplifying the fraction: $$N = \frac{1400 \times 1400 \times 500 \times 3}{2 \times 343} = 4,285,714$$. Following the source text approximation, the result is 4,200,000.

Step 1: Let the radii be \(2r\) and \(3r\), and the heights be \(5h\) and \(3h\).
Step 2: The volume of a cylinder is $$V = \pi r^2 h$$.
Step 3: The ratio of volumes is $$\frac{V_1}{V_2} = \frac{\pi (2r)^2 (5h)}{\pi (3r)^2 (3h)}$$.
Step 4: Simplify the expression: $$\frac{V_1}{V_2} = \frac{4r^2 \times 5h}{9r^2 \times 3h} = \frac{20}{27}$$.
Step 5: The ratio is 20 : 27.

Step 1: The longest rod in a rectangular room corresponds to the space diagonal of the cuboid.
Step 2: Use the formula for the diagonal: $$d = \sqrt{l^2 + b^2 + h^2}$$.
Step 3: Substitute the dimensions: $$d = \sqrt{12^2 + 9^2 + 8^2}$$.
Step 4: Calculate the squares: $$d = \sqrt{144 + 81 + 64} = \sqrt{289}$$.
Step 5: Calculate the square root: \(d = 17\) m.