Question

A codebreaking device is made up of a rectangular box filled with x cylinders of ball bearings placed together such that the diameter of the bearings and the cylinders are equal, and the cylinders line up evenly, touching, with no extra room inside the device. If the cylinders are the same height as the box, and the box is 18 inches long and 10 inches wide, what's the value of x?
(1) 9 cylinders can line up along the length of the box.
(2) Each ball bearing has a radius of 1.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient

Hint:

In Data Sufficiency, identify the "missing piece" of information needed to solve the problem (in this case, the cylinder's diameter). Then, evaluate each statement to see if it provides that missing piece.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is a Data Sufficiency problem involving geometry. We need to find the total number of cylinders (\(x\)) that can be packed into a rectangular box. The key is to find the diameter of a single cylinder.
Step 2: Key Formula or Approach:
Let \(D\) be the diameter of one cylinder.
The box dimensions are 18 inches (length) by 10 inches (width).
Number of cylinders along the length (\(N_L\)) = \( \frac{\text{Box Length}}{D} = \frac{18}{D} \).
Number of cylinders along the width (\(N_W\)) = \( \frac{\text{Box Width}}{D} = \frac{10}{D} \).
Total number of cylinders, \(x = N_L \times N_W\).
To find \(x\), we need to determine the value of \(D\).
Step 3: Detailed Explanation:
Analyzing Statement (1):
"9 cylinders can line up along the length of the box."
This gives us the value of \(N_L\).
\[ N_L = 9 \] Using our formula for \(N_L\):
\[ 9 = \frac{18}{D} \] We can solve for \(D\):
\[ D = \frac{18}{9} = 2 \text{ inches} \] Since we have found a unique value for the diameter \(D\), we can calculate the number of cylinders along the width (\(N_W = 10/2 = 5\)) and then the total number of cylinders (\(x = 9 \times 5 = 45\)).
Because we can find a unique value for \(x\), statement (1) is sufficient.
Analyzing Statement (2):
"Each ball bearing has a radius of 1."
The problem states that the diameter of the bearings and the cylinders are equal.
Radius = 1 inch.
Diameter \(D = 2 \times \text{Radius} = 2 \times 1 = 2\) inches.
This statement directly gives us the value of the diameter \(D\). Just as in statement (1), once we know \(D\), we can find a unique value for \(x\).
Therefore, statement (2) is sufficient.
Step 4: Final Answer:
Each statement alone is sufficient to determine the value of x.