
1. C
The formula for the length of the diagonal of a rectangular solid is where l, w, and h are the dimensions of the solid. Substitute 2l, 2w, and 2h for these values:
The length of the diagonal doubles, just like the rectangular solid’s other dimensions.
2. D
The volume of a cylinder is given by the formula . In the case of this cylinder, the radius is equal to height (r = h), so
Since we have a value only for the surface area of this cylinder, to find r, we must use the surface area formula, which is Because r = h in this cylinder, the surface area of this cylinder is
With the information given by the question, we can find the radius by setting either 4πr^{2} = 100π or r^{2} = 100π⁄4π = 25. Either way, the radius of the cylinder works out to 5, which means that the volume of the cylinder is π5^{3} = 125π.
3. C
The formula for a cone’s surface area is πr^{2} + πrl.
A cone’s volume is
then its surface area multiplies by the square of that factor. If the dimensions of a cone are multiplied by the same factor, the volume becomes multiplied by the cube of that factor:
In general, for solids, if each dimension of a cone is multiplied by the same factor, the solid’s surface area is multiplied by the square of that factor, and its volume increases by the cube of that factor. If the surface area of Cone A doubles, its dimensions are multiplied by a factor of Thus, the cone’s volume is multiplied by a factor of Cone A’s new volume is
4. E
If the rectangle is rotated about a side of length 4, then the height of the cylinder will be 4 and the radius will be 6.
Once you visualize the cylinder, you can plug in the values for the volume of a cylinder:
