First, find the volume of the larger box: V_large = 10³ = 10 × 10 × 10 = 1,000 cubic inches. Next, find the volume of the smaller box: V_small = 6³ = 6 × 6 × 6 = 216 cubic inches. The volume of the packing material is the difference between the two volumes: 1,000 - 216 = 784 cubic inches.

Let the edge length of the small cube be 's'. Its volume is s³. The edge length of the large cube is '3s'. Its volume is (3s)³ = 3s × 3s × 3s = 27s³. To find how many times larger the volume is, divide the large volume by the small volume: (27s³) / s³ = 27. The large cube's volume is 27 times that of the small cube.

First, determine the dimensions of the water volume. The length is 30 cm. The width is half the length, so it is 30 cm / 2 = 15 cm. The height of the water is given as 10 cm. The volume of a rectangular prism is length × width × height. So, the volume of the water is 30 cm × 15 cm × 10 cm = 4,500 cubic cm.

To calculate the volume in cubic centimeters, all dimensions must be in centimeters. The length is given as 2 meters. Since 1 meter = 100 centimeters, 2 meters = 200 centimeters. Now, multiply the dimensions: Volume = 200 cm × 50 cm × 20 cm = 10,000 × 20 = 200,000 cubic cm.

To find the volume of the L-shaped block, calculate the volume of the original large prism and subtract the volume of the removed section. Volume of the large prism = 10 m × 4 m × 3 m = 120 cubic m. Volume of the removed section = 6 m × 2 m × 3 m = 36 cubic m. The volume of the remaining block is 120 - 36 = 84 cubic m.

Calculate the volume of the cone and the hemisphere separately, then add them. V_cone = \(\frac{1}{3}\pi(3^2)(10) = \frac{1}{3}\pi(9)(10) = 30\pi\) cubic cm. A hemisphere is half a sphere. V_sphere = \(\frac{4}{3}\pi(3^3) = \frac{4}{3}\pi(27) = 36\pi\). V_hemisphere = \(\frac{1}{2} \times 36\pi = 18\pi\) cubic cm. Total volume = V_cone + V_hemisphere = 30π + 18π = 48π cubic cm.

The volume of a prism is the area of its base multiplied by its height. The base is a right triangle, so its area is (1/2) × base × height = (1/2) × 6 cm × 8 cm = 24 square cm. The height of the prism is 12 cm. Therefore, the volume of the prism is 24 cm² × 12 cm = 288 cubic cm.

When you encounter a problem involving removing material from a solid shape, you need to calculate the volume of the original shape minus the volume of the removed material.

Start with the cube's volume: $$20^3 = 8,000$$ cubic cm.

Next, find the volume of the cylindrical hole. The diameter is 10 cm, so the radius is 5 cm. The cylinder goes through the entire cube, so its height equals the cube's side length (20 cm). Using the given value π = 3:

$$V_{cylinder} = πr^2h = 3 × 5^2 × 20 = 3 × 25 × 20 = 1,500$$ cubic cm

The remaining concrete volume is: $$8,000 - 1,500 = 6,500$$ cubic cm, which is answer D.

Looking at the wrong answers: A) 2,000 represents a major calculation error, possibly confusing dimensions or formulas entirely. B) 6,000 likely comes from incorrectly calculating the cylinder's volume as 2,000 instead of 1,500 - perhaps using the diameter (10) instead of radius (5) in the area calculation. C) 7,500 suggests subtracting only 500 from the cube's volume, which might result from miscalculating the cylinder's height or making an arithmetic error in the volume formula.

Remember that "drilling through" means the hole goes completely through the object, so the cylinder's height equals the full dimension of the shape being drilled. Always double-check whether you're using radius or diameter in circular calculations - this is a frequent source of errors.

The volume of a prism is V = Base Area × height. The volume of a pyramid is V = (1/3) × Base Area × height. Since the base area and height are the same for both shapes, the volume of the pyramid is exactly 1/3 of the volume of the prism. Therefore, the volume of the pyramid is (1/3) × 36 cubic meters = 12 cubic meters.

Substitute the radius r = 6 cm into the volume formula for a sphere. V = \(\frac{4}{3}\pi(6)^3\). First, calculate 6³ = 6 × 6 × 6 = 216. Now, substitute this back into the formula: V = \(\frac{4}{3}\pi(216)\). To simplify, divide 216 by 3, which is 72. Then, multiply by 4: V = 4π(72) = 288π cubic cm.