Two solids are similar when they have the same shape and their corresponding dimensions are proportional. This means that all angles are equal and all lengths maintain a constant ratio.

When the scale factor between corresponding lengths is 2, the surface area scales by 2², which equals 4. Thus, the larger solid's surface area is four times that of the smaller solid.

Volume scales with the cube of the linear scale factor, so when lengths are scaled by 3, volumes are scaled by 3³ = 27. This means the larger solid's volume is 27 times that of the smaller one.

A scale factor describes the proportional relationship between corresponding linear measurements of similar solids. It is used to determine how much larger or smaller one solid is in comparison to another.

The larger cube has an edge length of 2 cm multiplied by the scale factor 3, which gives 6 cm. Its volume is then calculated as 6³ = 216 cubic centimeters.

Since volume scales with the cube of the linear scale factor, the volume of the larger pyramid is 10 × 5³, which equals 1250 cubic units. This emphasizes the exponential increase in volume with scaling.

The scale factor from the smaller to the larger cylinder is 3/2. Since surface area scales as the square of the linear scale factor, the larger cylinder's surface area is 50 × (3/2)² = 112.5 cm². This illustrates the quadratic effect of scaling on surface area.

The scale factor is 7/4, so the surface area scales by (7/4)² = 49/16. Multiplying 96 by 49/16 gives 294 square units. This demonstrates the squared relationship between linear dimensions and surface area.

The volume of a cone is proportional to the cube of its linear dimensions, so the volume scale factor is (5/3)³ = 125/27. This answer quantifies the significant increase in volume due to scaling.

Since volume scales with the cube of the linear dimensions, the volume ratio is 1:(√2)³, which simplifies to 1:2√2. This demonstrates the exponential effect of scaling on volume.

Doubling the linear dimensions increases the surface area by 2² = 4 times and the volume by 2³ = 8 times. This question highlights the different scaling effects on area and volume.

Taking the cube root of the volume ratio 8:27 gives the linear scale factor of 2:3. This means that the corresponding heights are in the ratio 2:3.

Since volume scales with the cube of the linear scale factor, the model's volume is 1/10³ (or 1/1000) of the actual building's volume. Dividing 1,000,000 by 1000 gives 1000 cubic feet.

The height of the larger cylinder is found by multiplying the smaller cylinder's height (10 cm) by the scale factor (4), resulting in 40 cm. This illustrates the direct proportionality between linear dimensions and the scale factor.

Since lateral areas scale as the square of the linear dimensions, the lateral area of the larger pyramid is 20 × 3² = 180 square units. This problem highlights the squared increase in area resulting from a linear scaling factor.

Both the sphere and the cylinder have volumes that scale with the cube of the linear scale factor. Thus, scaling each by 3 increases their individual volumes by 3³, which is 27, and the total volume is also multiplied by 27.

The lateral surface area of similar solids scales with the square of the linear dimensions. With a dimension ratio of 5:8, the lateral surface area ratio becomes (5²):(8²), which simplifies to 25:64.

Surface area scales as the square of the linear scale factor. With a scale factor of 1:20, the model's surface area is 1/20² or 1/400 of the statue's surface area. Thus, 800 divided by 400 equals 2 square meters.

Taking the cube root of the volume ratio 125:512 yields the linear scale factor. Since the cube root of 125 is 5 and that of 512 is 8, the corresponding lengths are in the ratio 5:8.

For a uniform material, mass is proportional to volume. Reducing the linear dimensions by 1/4 reduces the volume - and thus the mass - by (1/4)³, which is 1/64.