A rectangular solid has a surface area of 94 square inches. What is it

Great (and tricky) question!

Target question: What is the volume of the rectangular solid?

This is a good candidate for rephrasing the target question.

Let x, y, and z be the dimensions (length, width and height) of the rectangular solid.
So, xyz is its volume

REPHRASED target question: What is the value of xyz?

Given: The rectangular solid has a surface area of 94 square inches
Each side of the rectangular solid is a RECTANGLE.
So, 2 sides have the dimensions x by y, 2 sides have the dimensions x by z, and 2 sides have the dimensions y by z
So, the total surface area = 2(xy + xz + yz) = 94
Divide both sides by 2 to get: xy + xz + yz = 47

Statement 1: The surface area of one side of the rectangular solid is 12.
Let's say this is the side with dimensions x by y
So, xy = 12
Is this enough information to determine the value of xyz? No.
We know nothing about the value of z.
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The surface area of one side of the rectangular solid is 20
Let's say this is the side with dimensions x by z
So, xz = 20
For the same reason as above, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that xy = 12
Statement 2 tells us that xz = 20
We also know that xy + xz + yz = 47

NOTE: since this is system of 3 equations with 3 variables, many people will conclude that we can solve it for x, y, and z. However, that rule only applies to linear equations. In this case, the equations are quadratic equations. So, let's see if we can solve this system and find the value of xyz

Take xy + xz + yz = 47 and replace xy with 12 and xz with 20
We get: 12 + 20 + yz = 47
So, yz = 15

We now have:
xy = 12, xz = 20 and yz = 15
This means that (xy)(xz)(yz) = (12)(20)(15)
Simplify: x²y²z² = 3600
Rewrite as: (xyz)² = 3600
So, xyz = 60....DONE!!!
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer:

ASIDE: Some people will say that, if (xyz)² = 3600, then EITHER xyz = 60 OR xyz = -60
However, we're dealing with a real world problem here, and in the real world, the volume cannot be negative. So, we can ignore the negative solution.


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