To find the length of the diagonal, we need the side length.

I) Gives us the volume of the cube. Take the cubed root to find the side length.

II) Gives us the surface area, divide by 6 (the number of faces in a cube) and take the square root to find the side length.

Diagonal of a cube is the side length times the square root of three. Alternatively, this can be found using the Pythagorean Theorem twice.

Either way, we can use I) or II) to find our side lenght and then our diagonal.

The diagonal of a rectangular prism is found via the formula 

The second statement reduces this to 

However, the actual length is unknown. Statement 1 allows the calculation of a numerical value:

Volume of a prism is found by:

We are given the volume in statement I.

We are told how the sides relate in statement II.

Put together these two statements will allow us to set up an equation to find the middle side.

A pirate wants to hide all of his treasure. He commisions a local woodworker to build him a series of wooden chests of volume of . Find the length of the longest side given the following:

I) The shortest side will be  the length of the medium side

II) The middle side will be 2 feet long

Use I) and II) to find the length of the smallest side

Next, use the short and medium sides, along with info in the prompt, to find the last side:

Find the height of a box used to ship a computer, given the following:

I) The computer, along with all the cushioning, will take up a space of 

II) The width of the box will be  the length of the box

Begin by recalling the volume of a rectangular prism formula:

Where l,w and h are length, width and height.

Next, use II) to set up a relationship between w and l

Then, use the voume formula:

As you can see, we still have two unknowns, and no way of finding either. 

Therefore, we do not have information.

Statement (1) informs us that the triangle is an equilateral triangle, with all sides equal to 6. Therefore, the height would divide the triangle into two 30-60-90 triangles, which have side lengths in a ratio of . For this triangle the hypotenuse would be 6 and the base would be 3. The height would therefore be . SUFFICIENT

Statement (2) also lets you deduce that the triangle is an equilateral triangle. Since the triangle is either isoscles or equilateral (at least 2 sides are equal), that means that two angles are equal. Therefore, if one angle is  , the other two also must be  . See above. SUFFICIENT

Since we don't know what type the triangle is, we would not only need information about the lengths of the side but also about the characteristics of the triangle. 

Statement 1 gives us the length of a side. However, we can't do anything, since we don't know the length of DC, which would allow us the know BD with the Pythagorean Theorem.

Statement 2 also only gives us information about one side of the triangle. Alone it doesn't allow us to calculate any other length.

Even taken together these statements are insufficient since, we don't know any pair of lengths to use in the Pythagorean Theorem. Even though the triangle looks like a isosceles triangle, it doesn't mean that it is.

To find the length DC, we need to know AD and AC or any of those two provided that ABC is isosceles or equilateral.

Statement 1 tells us the area of the triangle with information about a part of side AC. Since we don't know properties of the triangle, these other lengths can vest many values, just AC can be 12, 24 or 48. Therefore we don't have enough information.

Statement 2 gives us information about angles of the triangle. From what we are told we can see that the triangle is isosceles. Indeed, we know that  since BD is the height. Therefore . Now, that we know that the triangle is isosceles, we know that AC must be 12, since D is the midpoint of AC. Therefore DC must be 6.

Hence, both statements taken together are sufficient.

Statement 1 alone proves that  by the Angle-Angle Postulate, but does not prove anything about the sides, which would be needed to answer this question. Statement 2 alone only gives a relationship between one side of each triangle; without any further information, this is insufficient.

The two statements together, however, present sufficient evidence. Statement 1 proves that the triangles are similar. Statement 2 gives the ratio of one side in  to the corresponding side in , so, as the triangles are similar, this ratio is shared by all three pairs of corresponding sides. Since

,

1.1 is this common ratio, and the ratio of the area of  to  is , making  the larger triangle.

Statement 1 alone gives a proportion between two pairs of corresponding sides of the triangles. This is not enough to prove the triangles similar without the third side proportionality (SSS Simiilarity statement) or the congruence of the included angles (SAS Similarity statement).

Statement 2 gives two congruencies between corresponding angles, which by the Angle-Angle statement is enough to prove the triangles similar.