ACT, Geometry – diagonal of a rectangular prism

Q: What is the (straight-line) distance between points A and B on the figure below?

rectangularprism2

Answer:  7

Explanation:  We are finding the length of the green segment below:

 

rectangularprism2wline

This requires the three-dimensional version of the distance formula:

  2014-09-12_225150

There is another way to do this, using a progression of two right triangles, with the green segment being the hypotenuse of the second triangle, but I find the above formula much simpler.  It is just an extension of the ‘usual’ (2-dimensional) distance formula we are so familiar with from Algebra and Geometry.

BUT if you want an even easier way, and I know you do, use my TI-84 Plus program ‘DIST3D’.

 

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Geometry – Volume

 

Q: What is the maximum number of boxes with dimensions 4”x5”x8” that can be packed into one large box with dimensions 20”x20”x24”?
Answer:  60

Explanation: First, decide that a whole number of small boxes will entirely fill the large box.  To do this, look at the dimensions and match them.  4″ fits into 20″ five times.  5″ fits into 20″ four times.  8″ fits into 24″ three times.  That means, you’ll get five rows of small boxes, four boxes per row, and three layers deep.  So, the quickest way to get the answer is to multiply 5x4x3 = 60.

BUT if you don’t notice that, just do (large volume)/(small volume).

Large volume = 20x20x24 = 9600

Small volume = 4x5x8 = 160

(large volume)/(small volume) = 9600/160 = 60

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Geometry – prism

 How many vertices does a rectangular prism have?


Answer:   8
rectangularprismvertices

If you have trouble drawing it or visualizing it, consider this instead:  A prism has two identical bases.  All of the vertices are along the edges of the two bases.  Since a rectangular prism has rectangles for bases, each base will have 4 vertices.  Two rectangular bases will have 8 vertices total.

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