The area of a rectangle is given by the expression 0.2x^2 + 0.5x, where x gt 0. The length of...

1. TRANSLATE the problem information

• Given information:
  • Area = \(0.2\mathrm{x}^2 + 0.5\mathrm{x}\)
  • Length = \(2\mathrm{x} + 5\)
  • Width = \(\mathrm{kx}\)
  • Need to find constant k

• What this tells us: We can use the rectangle area formula to set up an equation.

2. INFER the approach

• Since we know the area and both dimensions of a rectangle, we can use \(\mathrm{Area} = \mathrm{Length} \times \mathrm{Width}\)
• This will give us an equation with k that we can solve

3. TRANSLATE the area relationship into algebra

Set up the equation: \(0.2\mathrm{x}^2 + 0.5\mathrm{x} = (2\mathrm{x} + 5)(\mathrm{kx})\)

4. SIMPLIFY the right side by expanding

• Use distributive property: \((2\mathrm{x} + 5)(\mathrm{kx}) = (2\mathrm{x})(\mathrm{kx}) + (5)(\mathrm{kx})\)
• This gives us: \(0.2\mathrm{x}^2 + 0.5\mathrm{x} = 2\mathrm{kx}^2 + 5\mathrm{kx}\)

5. INFER how to solve for k

• Since both sides are polynomials in x, corresponding coefficients must be equal
• This means: coefficient of \(\mathrm{x}^2\) on left = coefficient of \(\mathrm{x}^2\) on right
• And: coefficient of x on left = coefficient of x on right

6. SIMPLIFY by equating coefficients

• From \(\mathrm{x}^2\) terms: \(0.2 = 2\mathrm{k}\) → \(\mathrm{k} = 0.2 \div 2 = 0.1\)
• From x terms: \(0.5 = 5\mathrm{k}\) → \(\mathrm{k} = 0.5 \div 5 = 0.1\)
• Both give the same result: \(\mathrm{k} = 0.1\)

Answer: B) 0.1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that "area equals length times width" needs to be set up as an equation with the given expressions.

Instead, they might try to work with the expressions separately or get confused about which expression represents what dimension. This leads to confusion and guessing rather than systematic equation setup.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(0.2\mathrm{x}^2 + 0.5\mathrm{x} = (2\mathrm{x} + 5)(\mathrm{kx})\) but make mistakes when expanding the right side.

They might incorrectly distribute and get something like \(2\mathrm{kx}^2 + 5\mathrm{x}\) instead of \(2\mathrm{kx}^2 + 5\mathrm{kx}\), missing that both terms on the right need to be multiplied by k. This leads to wrong coefficient equations and selecting an incorrect answer choice.

The Bottom Line:

This problem tests whether students can bridge geometry (rectangle area) with algebra (polynomial expressions). The key insight is recognizing that setting geometric relationships equal creates solvable algebraic equations.