Question

Find the percentage change in the volume of a cylinder.
1. The diameter is increased by 20%.
2. The height is increased by 21%.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question ask
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Hint:

For percentage change problems with formulas involving products of variables (like V = πr²h), you can analyze the multipliers. A 20% increase is a multiplier of 1.2. A 21% increase is a multiplier of 1.21. The new volume becomes \(V_{new} = \pi (1.2r)^2 (1.21h) = (1.2^2 \times 1.21) V_{old} = (1.44 \times 1.21) V_{old}\). This method is faster than writing out the full percentage change formula.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The volume of a cylinder depends on two variables: its radius (or diameter) and its height. To find the percentage change in volume, we need to know how both of these dimensions have changed.
Step 2: Key Formula or Approach:
The volume \( V \) of a cylinder is given by \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height.
The percentage change is calculated as:
\[ \text{Percentage Change} = \frac{\text{New Volume} - \text{Original Volume}}{\text{Original Volume}} \times 100% \] Step 3: Detailed Explanation:
Let the original radius be \( r_1 \), original height be \( h_1 \), and original volume be \( V_1 = \pi r_1^2 h_1 \).
Let the new radius be \( r_2 \), new height be \( h_2 \), and new volume be \( V_2 = \pi r_2^2 h_2 \).
Analyzing Statement (1): The diameter is increased by 20%.
An increase in diameter by 20% means the new diameter is \( 1.2 \) times the old one. Since the radius is half the diameter, the new radius \( r_2 \) is also \( 1.2 \) times the old radius \( r_1 \). So, \( r_2 = 1.2 r_1 \).
The new volume would be \( V_2 = \pi (1.2 r_1)^2 h_2 = \pi (1.44 r_1^2) h_2 \).
The percentage change would be \( \frac{1.44 \pi r_1^2 h_2 - \pi r_1^2 h_1}{\pi r_1^2 h_1} \). This simplifies to \( 1.44 \frac{h_2}{h_1} - 1 \).
Since we don't know the relationship between \( h_2 \) and \( h_1 \), we cannot find the percentage change. Statement (1) is not sufficient.
Analyzing Statement (2): The height is increased by 21%.
This means the new height \( h_2 \) is \( 1.21 \) times the old height \( h_1 \). So, \( h_2 = 1.21 h_1 \).
The new volume would be \( V_2 = \pi r_2^2 (1.21 h_1) \).
The percentage change depends on the change in radius (\( r_2 \)), which is unknown. Statement (2) is not sufficient.
Analyzing Statements (1) and (2) Together:
From statement (1), we have \( r_2 = 1.2 r_1 \).
From statement (2), we have \( h_2 = 1.21 h_1 \).
Now we can express the new volume \( V_2 \) in terms of the original volume \( V_1 \):
\[ V_2 = \pi r_2^2 h_2 = \pi (1.2 r_1)^2 (1.21 h_1) \] \[ V_2 = \pi (1.44 r_1^2) (1.21 h_1) \] \[ V_2 = 1.44 \times 1.21 \times (\pi r_1^2 h_1) \] \[ V_2 = 1.7424 V_1 \] Now we can calculate the percentage change:
\[ \text{Percentage Change} = \frac{V_2 - V_1}{V_1} \times 100% = \frac{1.7424 V_1 - V_1}{V_1} \times 100% \] \[ = \frac{0.7424 V_1}{V_1} \times 100% = 74.24% \] Since we can find a unique value for the percentage change, the two statements together are sufficient.
Step 4: Final Answer:
Neither statement alone is sufficient, but together they are sufficient.