The volume of a rectangular solid (rectangular prism) is to be increased by 50 percent without changing the area of its base. By what percentage must the height of the solid be increased?

Introduction / Context:
This question tests understanding of how the dimensions of a prism relate to its volume. A rectangular solid has a base area and a height, and the volume is the product of these two quantities. If the base area is kept constant, any change in volume must come entirely from changing the height. The problem asks how much the height must increase to increase volume by 50 percent under this condition.

Given Data / Assumptions:

• Let base area be B (constant).
• Let original height be h.
• Original volume V = B * h.
• New volume is 50 percent greater than original, so V_new = 1.5 * V.
• Base area does not change.

Concept / Approach:
Since V = B * h and B stays the same, any percentage change in volume must be equal to the percentage change in height. If V_new = B * h_new, then:
B * h_new = 1.5 * B * h. The base area B cancels out, leaving a simple proportion between h_new and h. From this, we can directly deduce the required percentage increase in height.

Step-by-Step Solution:
Step 1: Express original and new volumes. V = B * h. V_new = 1.5 * V = 1.5 * B * h. Step 2: Write new volume in terms of new height. V_new = B * h_new. Step 3: Equate the two expressions for V_new. B * h_new = 1.5 * B * h. Step 4: Cancel B on both sides. h_new = 1.5 * h. Step 5: The new height is 1.5 times the original height. Step 6: Percentage increase in height = (1.5 - 1) * 100% = 0.5 * 100% = 50%.

Verification / Alternative check:
Take a simple example: base area B = 10 cm^2 and original height h = 10 cm. Then original volume V = 10 * 10 = 100 cm^3. A 50 percent increase in volume means V_new = 150 cm^3. If we keep base area at 10 cm^2, the new height must be h_new = 150 / 10 = 15 cm. This is 5 cm more than the original height, which is a 50 percent increase (5 is half of 10).

Why Other Options Are Wrong:
40%, 30% and 20% increases in height would produce smaller percentage increases in volume because the base area is constant. For example, a 20 percent increase in height would yield only a 20 percent increase in volume.
10% is clearly too small; it would yield a volume of 110 percent of the original, not 150 percent.

Common Pitfalls:
Some learners mistakenly think that the volume increase will be more complicated because volume is three dimensional. However, when the base area is fixed, volume is directly proportional to height. Another common misunderstanding is to think that part of the volume change might be absorbed by the base, but the problem explicitly states that the base is unchanged. Focusing on the direct proportionality between volume and height with constant base area avoids these errors.

Final Answer:
The height of the solid must be increased by 50%.