Bobs Math Question: The Official Answers
EDIT: Please note: This is a single post explaining the answer to a question posted earlier on this blog.
This site is NOT intended as a general purpose site in which to get help with your math homework.
If you're having problems with your math homework, then you should consider asking your parents for help, you're not likely to find it here, sorry about that.
Ok, he's back :) My last post was a math problem that the teacher in my wife's classroom gave to the students (mostly 11 and 12 year olds fwiw).
Here's the official answer to the problem, the kids needed to show ALL the calculations (sorry for the word-junk):
Pyramid L=W=2’ H2 = 22 – 12 so H = 1.73
V =1/3*l*w*h
= 1/3*2*2*1.73 = 2.31 cubic feet
SA =b2 + 2bh
= (2)2 + 2*(2)*1.73
= 4 + 6.92 = 10.92 square feet.
Triangles
V=B*h SA = front + back + 3 sides
= 2*(1/2*l*h) + 3* L*W
Triangle #1 : L=8’, W=2’ H2 = 82 – 42 H = 6.93
V = 1/2*8*6.93*2 = 55.44 cubic feet
SA = 2(1/2*8*6.93) + 3*8*2 = 103.44 square feet
Triangle #2 : L=9’, W=2’ H2 = 92 – 4.52 H = 7.79
V = 1/2*9*7.79*2 = 70.11 cubic feet
SA = 2(1/2*9*7.79) + 3*9*2 = 124.11 square feet
Triangle #3 : L=10’, W=2’ H2 = 102 – 52 H = 8.66
V = 1/2*10*8.66*2 = 86.6 cubic feet
SA = 2(1/2*10*8.66) + 3*10*2 = 146.6 square feet
Base of Tree: L=W=2’ H= 3’
V = L*W*H = 2*2*3 = 12 cubic feet
SA = 2(L*H) + 2(W*H) + 2(L*W)
= 2(2*3 + 2*3 + 2*2)
= 2(6 + 6 + 4)
= 32 square feet
6 cones with H=1’, R=.5’, S= 1.12’
V = 1/3*π*r2h = 1/3 * 3.14*.52 * 1 = .26 cubic feet
Total volume = 6*.26 = 1.56 cubic feet
Volume before cutouts:
Pyramid 2.31
Triangle #1 55.44
Triangle #2 70.11
Triangle #3 86.60
Base 12.00
Cones 1.56
TOTAL 228.02
Cubic feet
Surface Area before cutouts:
Pyramid 10.92
Triangle #1 103.44
Triangle #2 124.11
Triangle #3 146.60
Base 32.00
Cones 15.30
TOTAL 432.37
Square
Cutout Calculations - Volume
All of the volume of the cutouts are subtracted from the total volume of the Christmas tree.
There are 6 cylinders total.
1 has r=1, h=2
4 have r=1.5, h=2
1 has r=2, h=2
V = πr2h SA = 2πr2 + 2πrh
V = π*(12 + 4(1.52) + 22)*2
= π*(1+9+4)*2
= 3.14*14*2 = 87.92 cubic feet
Small Triangular Prisms
There are three triangular prisms.
1 has L=B=1 and W = 2’
H2 = 12 - .52 so H= .87’
2 have L=B=1.5 and W = 2
H2 = 1.52 - .752 so H = 1.69’
V = Bw where B=1/2*l*h
V = (1/2*1*.87*2) + 2*(1/2*1.5*1.69*2)
= .87 + 5.07
= 5.94 cubic feet
Total volume to subtract:
87.92
+5.94
93.86 cubic feet
Christmas tree volume minus cutouts:
228.02
-93.86
134.16 Cubic Feet total
Cutout Calculations – SA
The front and back SA’s are subtracted from the total SA of the Christmas Tree but the side SA’s are added to the total.
Cylinders
Front and back SA = 2πr2
Side SA = 2πrh
Front and Back SA
= 2π(12 + 4*1.52 + 22)
=6.28 * (1+9+4)
= 87.92 Square feet
Side SA
= 2πrh
=2*π*(1+4*1.5+2)*2
= 12.56 * 9 = 113.04 Square feet
Small Triangular Prisms
Front and Back SA
= 2*1/2*b*h
= b*h
= 1*.87 + 2(1.5*1.69)
= .87 + 5.07
= 5.91 Square Feet
Side SA
= 3*b*w
= 3*(1+1.5+1.5)*2
= 24 square feet
Twice the SA of top of Base
=2(2*2)=8 Square Feet
SA to Add: 137.04
SA to Subtract: 101.83
Total SA to add: 35.21
Christmas Tree SA plus cutouts:
432.37
+35.21
467.58 Square Feet Total
Edit: Reduced Google juice of this post by changing the title from "Bobs Math Answers" to something more accurate - this post isn't intended to be a Q&A for students who are having trouble with their math homework :)