math summer work?

Question

The volume of a certain rectangular solid is 27 cubic centimeters. Its total surface area is 162 square centimeters, and its three dimensions are in geometric progression. What is the sum of the lengths in centimeters of all the edges of this solid?

please show me work if you can get this, someone tried to explain this to me but i couldn't understand it. Thanks

Answer 1

OK first lets put together all details:

Volume = l * w * h = 27 cm^3
Surface Area = 2 (l*w + l*h + h*w) = 162 cm^2
So l*w + l*h + h*w = 162/2 cm^2 = 81 cm^2


Thats about it, since I don't really understand geometric progressions, sorry...

Answer 2

Let's write down the given conditions.
Volume = l * w * h = 27
Surface Area = 2 (l*w + l*h + h*w) = 162
So l*w + l*h + h*w = 162/2 = 81
Now l = l, w = rl, h = r²l, since l,w and h are in geometric
progression.
So r³l³ = 27
rl = 3.
The second condition yields
l²r + l²r² + l²r³ = 81
l²r(1+r+r²) = 81
l *rl(1+r+r²) = 81
Since rl = 3 we get
3l(1+r+r²)=81
l(1+r+r²)=27
lr=3.
So
(1+r+r²)/r = 9
1+r+r² = 9r
r²-8r+1= 0.
Now you have a simple quadratic which you can
solve for r.
Once you have r, we know
l = r/3
w = 3
h = 3r.
Note that there are 2 values of r which work here.
I'll let you finish the job!

Answer 3

Sorry, 162 is wrong, it should be 78.
l*b*h = 27
2(l*b + b*h + h*l) = 78
=> l*b + b*h + h*l = 39
also h, b, l are in GP
=> l*h = b^2
then
l*b*h = 27 => b^3 = 3^3 => b = 3
l*h = b^2 = 3^2 = 9
l*b + b*h + h*l = 39
=> b(l + h) + 9 = 39
=> 3(l + h) = 30
=> l + h = 10 and l*h = 9
giving h = 1 and l = 9
then sum of all its edges
= 4(l + b + h)
= 4(9 + 3 + 1)
= 52 (in centimetres or what you like)

Answer 4

Let's call the three edges A, B and C.

The volume is just ABC, so ABC=27.

The surface area is 2(AB+AC+BC), so that equals 162.

Finally, "its three dimensions are in geometric progression" means that you multiply the smallest by some number to get the next largest, and you mutiply that by the same number to get the next one. In equations:
B = pA
C = pB = p^2A
for some fixed p.

You can use these to rewrite the volume and surface area equations in term of A and p.

For example, the volume is ABC = A * pA * p^2A = (pA)^3
=> (pA)^3 = 27
=> pA = 3

Do the same to the surface area equation and you should be a lot closer to the answer you need.

Answer 5

I dont't believe this problem has a solution.