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A rectangular prism has a volume of \(720cm^3\) and a surface area of \(666cm^3\). Its edges are integers. What are the side lengths of the prism?

 

I solved this problem the "dumb" way, by trying out every case, but I wonder if there is a "better", more algebraic way to solve it. 

 

Thanks for any help!!

 Apr 17, 2022
 #1
avatar+55 
0

\(𝑥𝑦𝑧=720 \\ 𝑥𝑦+𝑥𝑧+𝑦𝑧=333 \),

Im not sure what to do from here, we could make \(𝑥≥𝑦≥𝑧\), but I don't see how that helps

 Apr 17, 2022
 #2
avatar+1351 
+1

I did that, and I just brute forced it from here...

 

Surely there's a better way...

BuilderBoi  Apr 17, 2022
 #4
avatar+55 
-2

Did It come with a answer key?

qjin27  Apr 17, 2022
 #6
avatar+1351 
+1

No, but I found the answer already by guessing and checking, \(x = 16\)\(y = 15\) and \(z = 3\)

BuilderBoi  Apr 17, 2022
 #7
avatar+55 
-2

I think because z=8 is the case that z is the largest, and z=1 is when z is at the smallest, so z has to be in between that

qjin27  Apr 17, 2022
 #9
avatar+117104 
0

It is a good start  laugh

Melody  Apr 18, 2022
 #3
avatar+55 
+1

I found online that you can do

 \(720≥z^3 \\ 1≤𝑧≤8 \\ \),

Then you try out 8 cases

 Apr 17, 2022
 #5
avatar+1351 
+1

That's interesting, I wonder why it works....

BuilderBoi  Apr 17, 2022
 #8
avatar+117104 
+3

Here is how I approached it

 

\(xyz=720\\ xy+xz+yz=333\\ \sqrt[3]{333}\approx 8.9\)

 

this means that at least one of the dimensions is 8 or less, and at least one is 9 or more

 

Factor  720 and you get    

\(720=2^4*3^2*5\)

the factors 8 and smaller are  1,2,3,4,5,6,8   

 

Try x=8 

If x=8 then  yz=720/8 = 90

8y+8z+90=333   so     8y+8z = 333-90 = 243   

But 243 is not divisable by 8 so 8 is not one ofthe dimensions.

 

Try x=6     I ran into the same problem

Try x=5      same problem

Try x=4    same problem

 

Try x=3

If x=3 then  yz= 720/3 = 240

3y+3z+240=333

3y+3z=93  so y+z=31

31=15+16, that sounds promising

 

If the sides, x,y and z  are  3, 15, and 16  then

(3*15)+(3*16)+(15*16) = 45+48+240 = 333  Bingo

 

the sides are   3,15 and 16

 Apr 18, 2022
 #10
avatar+1351 
+1

Thanks for answering everyone!!

 Apr 18, 2022

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