One side of a flat board has an area of 874 square inches. If a pressure of 19 pounds per...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{Area} = 874\text{ square inches}\)
  • \(\mathrm{Pressure} = 19\text{ pounds per square inch}\)
  • Need to find: Total force in pounds

2. INFER the mathematical relationship needed

• We need to connect pressure, force, and area
• Since pressure is 'force per unit area,' we can use: \(\mathrm{Pressure} = \frac{\mathrm{Force}}{\mathrm{Area}}\)
• To find force, we rearrange this to: \(\mathrm{Force} = \mathrm{Pressure} \times \mathrm{Area}\)

3. SIMPLIFY by substituting and calculating

• \(\mathrm{Force} = 19\text{ pounds/square inch} \times 874\text{ square inches}\)
• \(\mathrm{Force} = 16,606\text{ pounds}\) (use calculator)
• Notice how the 'square inches' units cancel out, leaving just 'pounds'

Answer: 16,606 pounds

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Students don't remember or understand the pressure formula \(\mathrm{P} = \frac{\mathrm{F}}{\mathrm{A}}\)

Without knowing this fundamental relationship, students may guess at how pressure, force, and area relate to each other. Some might try dividing the area by the pressure, or adding them together, leading to incorrect calculations and confusion about what operation to perform.

Second Most Common Error:

Weak INFER reasoning: Students know pressure involves force and area but can't figure out how to rearrange the formula

They might recognize that \(\mathrm{P} = \frac{\mathrm{F}}{\mathrm{A}}\) is relevant but get confused about whether to multiply or divide the given values. This algebraic manipulation challenge can cause them to select an incorrect approach and arrive at the wrong numerical answer.

The Bottom Line:

This problem tests whether students can connect a real-world physics concept (pressure) to its mathematical formula and perform the necessary algebraic rearrangement. Success requires both conceptual knowledge of the pressure formula and the process skill to manipulate it correctly.