\(\mathrm{c(x) = mx + 500}\)A company's total cost \(\mathrm{c(x)}\), in dollars, to produce x shirts is given by the function...
GMAT Algebra : (Alg) Questions
\(\mathrm{c(x) = mx + 500}\)
A company's total cost \(\mathrm{c(x)}\), in dollars, to produce \(\mathrm{x}\) shirts is given by the function above, where \(\mathrm{m}\) is a constant and \(\mathrm{x \gt 0}\). The total cost to produce \(\mathrm{100}\) shirts is \(\mathrm{\$800}\). What is the total cost, in dollars, to produce \(\mathrm{1000}\) shirts? (Disregard the $ sign when gridding your answer.)
1. TRANSLATE the problem information
- Given information:
- Cost function: \(\mathrm{c(x) = mx + 500}\)
- Specific cost: 100 shirts cost $800
- Unknown: cost for 1000 shirts
- What this tells us: We need to find the value of m before we can answer the main question
2. INFER the solution strategy
- We can't directly find \(\mathrm{c(1000)}\) because we don't know what m is
- Strategy: Use the given information (100 shirts = $800) to find m first
- Then use the complete function to find \(\mathrm{c(1000)}\)
3. SIMPLIFY to find the unknown parameter m
- Substitute the known values into the function:
\(\mathrm{c(100) = 800}\)
\(\mathrm{m(100) + 500 = 800}\) - Solve for m:
\(\mathrm{100m = 800 - 500}\)
\(\mathrm{100m = 300}\)
\(\mathrm{m = 3}\)
4. SIMPLIFY to find the final answer
- Now we have the complete function: \(\mathrm{c(x) = 3x + 500}\)
- Substitute x = 1000:
\(\mathrm{c(1000) = 3(1000) + 500}\)
\(\mathrm{c(1000) = 3000 + 500}\)
\(\mathrm{c(1000) = 3500}\)
Answer: 3500
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students try to jump directly to finding \(\mathrm{c(1000)}\) without realizing they need to find m first.
They might attempt to substitute x = 1000 into \(\mathrm{c(x) = mx + 500}\), getting \(\mathrm{c(1000) = 1000m + 500}\), but then realize they don't know what m is. This leads to confusion and guessing, or they might incorrectly assume \(\mathrm{m = 1}\) and get 1500 as their answer.
Second Most Common Error:
Poor SIMPLIFY execution: Students set up the equation correctly but make arithmetic errors.
For example, when solving \(\mathrm{100m + 500 = 800}\), they might subtract incorrectly and get \(\mathrm{100m = 200}\) instead of \(\mathrm{100m = 300}\), leading to \(\mathrm{m = 2}\). This would give \(\mathrm{c(1000) = 2(1000) + 500 = 2500}\) instead of the correct 3500.
The Bottom Line:
This problem tests whether students understand that unknown parameters in functions must be determined before the function can be fully used. The key insight is recognizing the two-step process rather than trying to solve everything at once.