A manufacturing company produces two sizes of cylindrical containers that each have a height of 50 centimeters. The radius of...

1. TRANSLATE the problem information

• Given information:
  • Both containers have height = 50 cm
  • Container A radius = 16 cm
  • Container B radius is "25% longer than" Container A's radius
  • Need volume of Container B

• What "25% longer than 16" means mathematically:
  \(\mathrm{16 + (0.25)(16)}\)

2. INFER the solution approach

• To find volume, we need the radius of Container B first
• Once we have Container B's radius, we can use the cylinder volume formula

3. SIMPLIFY to find Container B's radius

Container B radius = \(\mathrm{16 + (0.25)(16)}\)
\(\mathrm{= 16 + 4}\)
\(\mathrm{= 20\text{ cm}}\)

4. SIMPLIFY to find Container B's volume

Using \(\mathrm{V = \pi r^2h}\):
\(\mathrm{V = \pi(20)^2(50)}\)
\(\mathrm{V = \pi(400)(50)}\)
\(\mathrm{V = 20{,}000\pi\text{ cubic centimeters}}\)

Answer: B. \(\mathrm{20{,}000\pi}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "25% longer than 16" as simply "25" instead of "\(\mathrm{16 + (0.25)(16) = 20}\)"

They calculate volume using radius = 25:
\(\mathrm{V = \pi(25)^2(50) = \pi(625)(50) = 31{,}250\pi}\)

This may lead them to select Choice D (\(\mathrm{31{,}250\pi}\))

Second Most Common Error:

Poor SIMPLIFY execution: Students find the correct radius (20) but make calculation errors in the volume formula

Common mistakes include:

• Using \(\mathrm{V = \pi(20)(50) = 1{,}000\pi}\) (forgetting to square the radius)
• Using \(\mathrm{V = \pi(16)(20) = 320\pi}\) (multiplying the two radii instead of using the formula)

This may lead them to select Choice A (\(\mathrm{16{,}000\pi}\)) or causes confusion and guessing

The Bottom Line:

This problem tests your ability to translate percentage language into mathematics and then carefully execute the cylinder volume formula. The key insight is recognizing that "25% longer than 16" means \(\mathrm{16 + 4 = 20}\), not just 25.