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

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

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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 :)