Two similar solids have a scale factor of 3:5. If the height of solid I is 3 cm, find the height of solid II.

To find the height of solid II, we can set up a proportion using the scale factor:

scale factor of I : scale factor of II = height of I : height of II

Since we know that the scale factor is 3:5, we can set up the proportion as follows:

3:5 = 3 cm : h

To solve for h, we can cross-multiply and then divide:

3 * h = 3 cm * 5
h = (3 cm * 5) / 3
h = 15 cm

Therefore, the height of solid II is 15 cm.

To find the surface area of solid II, we need to use the scale factor to determine the relationship between the surface areas of the two solids.

The relationship between the surface areas of two similar solids is given by the square of the scale factor.

So, if the scale factor between the two solids is 3:5, then the ratio of their surface areas is (3/5)^2.

Let's calculate the surface area of solid II using this ratio:

surface area of II = (3/5)^2 * surface area of I

Given that the surface area of I is 54π cm^2, we can substitute it into the equation:

surface area of II = (3/5)^2 * 54π cm^2

Calculating this expression:

surface area of II = (9/25) * 54π cm^2
surface area of II = 194.4π cm^2

Therefore, the surface area of solid II is 194.4π cm^2.

To find the volume of solid I, we need to use the scale factor to determine the relationship between the volumes of the two solids.

The relationship between the volumes of two similar solids is given by the cube of the scale factor.

So, if the scale factor between the two solids is 3:5, then the ratio of their volumes is (3/5)^3.

Let's calculate the volume of solid I using this ratio:

volume of I = (3/5)^3 * volume of II

Given that the volume of II is 250π cm^3, we can substitute it into the equation:

volume of I = (3/5)^3 * 250π cm^3

Calculating this expression:

volume of I = (27/125) * 250π cm^3
volume of I = 54π cm^3

Therefore, the volume of solid I is 54π cm^3.