AP Biology: Unit 2, Cell Size & Surface Area (Drill 7)

Question 1. Based on the data in the table, which of the following correctly describes the relationship between cell radius and SA:V ratio?

• A) As radius doubles, the SA:V ratio also doubles, because surface area and volume scale up together.
• B) As radius doubles, SA:V ratio remains constant because both surface area and volume increase.
• C) As radius doubles, SA:V ratio is halved, because volume increases as a cube of the radius while surface area increases as a square. ✓
• D) As radius doubles, SA:V ratio increases because larger cells have more total surface area.

Explanation: The data show SA:V ratios of 3.0, 1.5, and 0.75 as radius doubles from 1 to 2 to 4 um -- each ratio is exactly half the previous. This reflects the mathematical relationship: surface area scales with r^2 while volume scales with r^3, so SA:V ratio scales as 1/r. Doubling the radius doubles the denominator of the ratio without a proportional increase in surface area. A and D are directly contradicted by the data. B is incorrect -- the ratio does not remain constant; it systematically decreases.

Question 2. A student calculates the SA:V ratio for a spherical cell with a radius of 3 um. Using the formulas provided, which of the following is closest to the correct SA:V ratio?

• A) 0.5
• B) 1.0 ✓
• C) 1.5
• D) 2.0

Explanation: Surface area = 4(pi)(3)^2 = 4(pi)(9) ~= 113.1 um^2. Volume = (4/3)(pi)(3)^3 = (4/3)(pi)(27) ~= 113.1 um^3 (the same as surface area at r = 3, due to the geometry at this radius). SA:V ratio = 113.1 / 113.1 = 1.0. This also follows the formula SA:V ratio = 3/r, so at r = 3, SA:V ratio = 3/3 = 1.0. A corresponds to r = 6. C corresponds to r = 2. D corresponds to r = 1.5.

Question 3. A researcher observes that intestinal epithelial cells have densely packed microvilli on their apical surface. Based on the passage, which of the following best explains the functional significance of this adaptation?

• A) Microvilli increase the overall volume of the cell, allowing it to store a much larger quantity of absorbed nutrients within its interior.
• B) Microvilli reduce the SA:V ratio of the cell, making transport across the membrane more efficient.
• C) Microvilli increase the surface area of the cell membrane without proportionally increasing cell volume, improving the cell's capacity for material exchange. ✓
• D) Microvilli allow the cell to divide more rapidly by providing additional membrane for daughter cells.

Explanation: The passage explicitly states that structural adaptations such as microvilli increase surface area without proportionally increasing volume -- directly addressing the SA:V ratio constraint. Microvilli do not increase cell volume (A). They increase, not decrease, the SA:V ratio (B). Cell division is unrelated to microvilli function in this context (D).

Question 4. Two cells are identical in all respects except that Cell X has a radius of 1 um and Cell Y has a radius of 4 um. Both cells have the same metabolic rate per unit volume. Which of the following best predicts the consequence for Cell Y?

• A) Cell Y will exchange materials more efficiently than Cell X because it has a greater total surface area, despite a lower SA:V ratio.
• B) Cell Y will accumulate waste products and deplete nutrients more rapidly than Cell X because its lower SA:V ratio means the membrane surface is insufficient to support its metabolic demands. ✓
• C) Cell Y and Cell X will exchange materials at the same rate because metabolic rate per unit volume is identical.
• D) Cell Y will compensate for its lower SA:V ratio by automatically reducing its own metabolic rate per unit volume to match the available membrane.

Explanation: Because Cell Y has a larger volume, it generates more total waste and requires more total nutrients -- but its SA:V ratio is only 0.75 compared to Cell X's 3.0, meaning it has far less membrane surface relative to its metabolic demand. This makes efficient exchange difficult and is precisely the constraint described in the passage. A confuses total surface area with SA:V ratio -- a larger total surface area does not help if volume has grown proportionally faster. C ignores the SA:V ratio relationship entirely. D introduces a compensatory mechanism not supported by the passage.

Question 5. A student claims that a cell could indefinitely increase in size as long as it continues to synthesize new membrane to expand its surface area. Which of the following best identifies the flaw in this reasoning?

• A) The claim is correct -- cells can grow indefinitely as long as membrane synthesis keeps pace with growth.
• B) The student ignores the fact that membrane synthesis simply stops once a cell reaches a certain fixed maximum size set by the genome.
• C) The student fails to account for the fact that volume increases faster than surface area as a cell grows, so even with continuous membrane synthesis, surface area cannot increase as quickly as volume, so the SA:V ratio will continue to decline and eventually become insufficient to support the cell's metabolic demands. ✓
• D) The claim is flawed because larger cells require less membrane surface area relative to their volume.

Explanation: The core flaw is mathematical: because volume scales as r^3 and surface area scales as r^2, a cell cannot synthesize membrane fast enough to maintain an adequate SA:V ratio as it grows larger. Even if membrane is continuously added, surface area cannot increase as quickly as volume, so the SA:V ratio still declines. A accepts the flawed premise. B is factually incorrect -- membrane synthesis does not have a fixed stopping point; the problem is the geometric relationship, not the rate of synthesis. D restates the consequence but does not identify the reasoning error.