All ISEE Upper Level Quantitative Resources
Example Questions
Example Question #1 : How To Find The Diagonal Of A Cube
What is the length of the diagonal of a cube with a side length of ? Round to the nearest hundreth.
It is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:
(Note that not all points are drawn in on the cube.)
The two points we are looking at are:
and
Solve this by using the distance formula. This is very easy since one point is all s. It is merely:
This is approximately .
Example Question #2 : How To Find The Diagonal Of A Cube
What is the length of the diagonal of a cube with a side length of ? Round to the nearest hundreth.
It is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:
(Note that not all points are drawn in on the cube.)
The two points we are looking at are:
and
Solve this by using the distance formula. This is very easy since one point is all s. It is merely:
This is approximately .
Example Question #3 : How To Find The Diagonal Of A Cube
What is the length of the diagonal of a cube with a volume of ? Round to the nearest hundredth.
First, you need to find the side length of this cube. We know that the volume is:
, where is the side length.
Therefore, based on our data, we can say:
Solving for by taking the cube-root of both sides, we get:
Now, it is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:
(Note that not all points are drawn in on the cube.)
The two points we are looking at are:
and
Solve this by using the distance formula. This is very easy since one point is all s. It is merely:
This is approximately .
Example Question #4 : How To Find The Diagonal Of A Cube
What is the length of the diagonal of a cube with a surface area of ? Round your answer to the nearest hundredth.
First, you need to find the side length of this cube. We know that the surface area is defined by:
, where is the side length. (This is because the cube is sides of equal area).
Therefore, based on our data, we can say:
Take the square root of both sides and get:
Now, it is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:
(Note that not all points are drawn in on the cube.)
The two points we are looking at are:
and
Solve this by using the distance formula. This is very easy since one point is all s. It is merely:
This is approximately .
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