What is the volume of a certain rectangular solid?
1. Two adjacent faces have areas 15 and 24 respectively.
2. Each of two opposite faces of the solid has area 40.
This is OG question.
The problem I am facing is understanding second statement.
Can anyone please help in understanding the meaning?
Two opposite faces of rectangular solid will always have same area. So does this statement mean that all 3 pairs of opposite faces have same areas?
Please help.
Target question:
What is the volume of a certain rectangular solid?
Aside: A rectangular solid is a box
Statement 1: Two adjacent faces of the solid have areas 15 and 24, respectively.
There are several different rectangular solids that meet this condition. Here are two:
Case a: the dimensions are 1x15x24, in which case
the volume is 360
Case b: the dimensions are 3x5x8, in which case
the volume is 120
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: Each of two opposite faces of the solid has area 40.
So, there are two opposite faces that each have area 40.
Definitely NOT SUFFICIENT
Statements 1 and 2 combined:
So, we know the area of each face (noted in blue on the diagram below).
Let's let x equal the length of one side.
Since the area of each face = (length)(width), we can express the other two dimensions in terms of x.
From here, we'll focus on the face that has area 40.
This face has dimensions (15/x) by (24/x)
Since the area is 40, we know that (15/x)(24/x) = 40
Expand: 360/(x^2) = 40
Simplify: 360 = 40x^2
Simplify: 9 = x^2
Solve: x = 3 or -3
Since the side lengths must be positive, we can be certain that
x = 3
When we plug x=3 into the other two dimensions, we get 15/3 and 24/3
So, the 3 dimensions are 3, 5, and 8, which means
the volume of the rectangular solid must be 120.
Since we
can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer =
C
Cheers,
Brent
