11 balls have radii X cm, X+1 cm, …, X+10 cm. Which of the following is true

about the volumes of these balls?

The mean is equal to the median.

The mean is greater than the median.The mean is less than the median.

The relative order of mean and median depends on the value of X.

The volumes of 11 balls with radii

X, X+1, …, X+10

cm follow the formula

V_i = \frac{4}{3}\pi r_i^3

, forming a right-skewed distribution due to the cubic growth. The mean is greater than the median. This holds true regardless of

X > 0

, as larger radii disproportionately inflate the average.

Volume Distribution Properties

The ordered volumes

V_1 < V_2 < … < V_{11}

are strictly increasing since the cube function amplifies differences at higher radii. With 11 values (odd count), the median is the 6th volume,

V_6 = \frac{4}{3}\pi (X+5)^3

. The mean is

\bar{V} = \frac{4}{3}\pi \cdot \frac{1}{11} \sum_{k=0}^{10} (X+k)^3

.

Expanding

(X+k)^3 = X^3 + 3X^2 k + 3X k^2 + k^3

, the sum becomes

11X^3 + 3X^2 S_1 + 3X S_2 + S_3

, where

S_1 = \sum k = 55

,

S_2 = \sum k^2 = 385

,

S_3 = \sum k^3 = 3025

. Thus,

\bar{V} = \frac{4}{3}\pi \left[ X^3 + \frac{3X^2 \cdot 5 + 3X \cdot 35 + 275}{11} \right] = \frac{4}{3}\pi \left[ X^3 + 15X^2 + 30X + 25 \right]

, or

\frac{4}{3}\pi (X+5)^3 + \frac{4}{3}\pi \cdot 30X

.

Why Mean Exceeds Median

Since

\bar{V} = V_6 + \frac{4}{3}\pi \cdot 30X

and

X > 0

,

\bar{V} > V_6

. The positive skew from cubed terms pulls the mean rightward, a hallmark of right-skewed data where mean > median.

Analysis of All Options

• The mean is equal to the median: Incorrect, as the skew term
  \frac{4}{3}\pi \cdot 30X > 0ensures inequality.
• The mean is greater than the median: Correct, proven algebraically for any
  X > 0.
• The mean is less than the median: Incorrect; this suits left-skewed data, opposite here.
• The relative order depends on X: Incorrect; the inequality is independent of X’s value (assuming positive radii).

This pattern aids statistics education, highlighting how nonlinear transformations like cubing create skewness in real-world data like spherical volumes.