1. Donatello starts with a marble cube of side length \(10.\) He then slices a pyramid off each corner, so that in the resulting polyhedron, all the edges have the same side length \(s.\) Find \(s.\)
Here's the image:
2. A rectangular prism has a total surface area of \(48.\) Also, the sum of all the edges of the prism is \(40.\) Find the length of the diagonal joining one corner of the prism to the opposite corner. (No image for #2.)
Thanks!
+23246 1) The total length of a side is 10.
The inner portion is s; this means that the left portion is 5 - ½s and the right portion is also 5 - ½s.
If you look at the top corner section (that is to be cut off), there is a right isosceles triangle at the top.
One side is 5 - ½s as is the other side. The hypotenuse is s.
Using the Pythagorean Theorem: (5 - ½s)2 + (5 - ½s)2 = s2
---> (25 - 5s + ¼s2) + (25 - 5s + ¼s2) = s2
---> 50 - 10s + ½s2 = s2
---> 100 - 20x + s2 = 2s2
---> s2 + 20s - 100 = 0
---> s = -10 + 10sqrt(2) (using the quadratic formula)
+23246 2) l = length h = height w = width
Total surface area: 2lw + 2lh + 2wh = 48 ---> lw + lh + wh = 24
Sum of all the edges: 4l + 4w + 4 h = 40 ---> l + w + h = 10
Let the first dimensions be: 4 + 2sqrt(2)
the second dimension: 4 - 2 sqrt(2)
the third dimension: 2
Check: Total Surface Area: (4 + 2sqrt(2)) · (4 - 2sqrt(2)) + (4 + sqrt(2))·2 + (4 - 2sqrt(2))·2
= (16 - 4) + (8 + 2sqrt(2)) + (8 - 2sqrt(2))
= 8 + 8 + 8 = 24
Check: Edges: (4 + 2sqrt(2)) + (4 - 2sqrt(2)) + 2 = 4 + 4 + 2 = 10
Found be guessing, and guessing, and guessing ...
