Space diagonal with a rectangular prism

Question

Mr. Bentley wants to ship his favorite math pencil to his buddy Arman. Mr. Bentley's pencil is 8 inches long. The only box that Mr.Bentley has to ship his pencil is a rectangular prism. The box is 4 inches long, 5 inches wide and fits the pencil exactly with no extra room. Express the height of the box in simplest radical form. The pencil in the rectangular prism goes from the top left ( first from the back) corner, and slants down into the bottom right of the back rectangle.

Answer 1

Mark,

Without a drawing, this is impossible to demonstrate. The equation, however, is very straight-forward. The diagonal through the center of a rectangular prism can be found by an extension of Pythagorean Theorem.

 

a² + b² + c² = d²  where d is the diagonal. So plugging in the given information we get:

 

4² + 5² + c² = 8²  OR  16 + 25 + c² = 64

 

c² = 23

 

c = √23 = height

 

I hope that helps.

Answer 2

The diagonal across the bottom of the box is √(4² + 5²) or √41.

This diagonal becomes the long leg of a right triangle. The short leg is the height of the box and the hypotenuse is the pencil.

8² = h² + (√41)²

64 = h² + 41

23 = h²

√23 = h