Question

Two spherical unicellular organisms, X and Y, have the same cytoplasmic composition, metabolic pathways, and live in identical nutrient-rich aerobic conditions. The diameters of X and Y are, respectively, 10 and 20 $\mu\text{m}$. When nutrients are not limiting, the ratio of total oxygen consumption per hour per cell of organisms Y to X is:

• A. 8:1
• B. 1:2
• C. 2:1
• D. 4:1

Hint:

For any three-dimensional scaling problem of identical composition, volume scales as the cube of the linear dimension ($d^3$). Thus, doubling the diameter ($2 \times$) increases the volume and overall metabolic capacity by $2^3 = 8$ times.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the ratio of total oxygen consumption per hour between two spherical unicellular organisms, X and Y, with diameters of $10\ \mu\text{m}$ and $20\ \mu\text{m}$ respectively, under non-limiting nutrient and aerobic conditions.

Step 2: Key Formula or Approach:
The volume of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 = \frac{1}{6} \pi d^3 \]
The oxygen consumption rate of a metabolically active cell under non-limiting aerobic conditions is proportional to its total metabolic demand, which scales directly with cell volume (cytoplasmic mass) rather than surface area.

Step 3: Detailed Explanation:

• We are given two spherical cells, X and Y, with identical cytoplasmic composition and metabolic pathways.

• The diameter of cell X is $d_X = 10\ \mu\text{m}$ and the diameter of cell Y is $d_Y = 20\ \mu\text{m}$.

• The volume of cell X is: \[ V_X = \frac{1}{6} \pi (10)^3 = \frac{1000\pi}{6}\ \mu\text{m}^3 \]

• The volume of cell Y is: \[ V_Y = \frac{1}{6} \pi (20)^3 = \frac{8000\pi}{6}\ \mu\text{m}^3 \]

• Since the cytoplasmic composition and metabolic pathways are identical, each unit volume of cytoplasm consumes oxygen at the same rate.

• Therefore, the total oxygen consumption per cell is directly proportional to the cell volume.

• The ratio of oxygen consumption of organism Y to organism X is: \[ \text{Ratio} = \frac{V_Y}{V_X} = \frac{8000}{1000} = 8:1 \]


Step 4: Final Answer:
Therefore, the ratio of total oxygen consumption per hour per cell of organisms Y to X is 8:1.