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Photographic film is manufactured by applying a very thin layer of a viscous solution, called a melt, to a transparent, inert support material. If \(25\) cubic centimeters of melt is applied per square meter of support material is an even layer, what is the thickness of the layer, in centimeters? (\(1 \text{ meter} = 100 \text{ centimeters}\))
Let's break down what's happening here in everyday terms. Imagine you're spreading a layer of honey on a slice of toast. If I tell you that you used exactly 25 cubic centimeters of honey and your toast has an area of 1 square meter, we want to figure out how thick that layer of honey is.
In this problem, we have:
The key insight is that when you spread a volume of material evenly over a surface, the thickness depends on how much material you have and how much area you're covering. More material or less area means thicker layer; less material or more area means thinner layer.
Process Skill: TRANSLATE - Converting the physical scenario into a mathematical relationship
Right now we have a problem: our volume is measured in cubic centimeters, but our area is measured in square meters. It's like trying to compare apples and oranges - we need everything in the same units.
Since the answer choices are all in centimeters, let's convert our area from square meters to square centimeters.
We know that \(1 \mathrm{meter} = 100 \mathrm{centimeters}\)
So \(1 \mathrm{square\,meter} = (100 \mathrm{cm}) \times (100 \mathrm{cm}) = 10,000 \mathrm{square\,centimeters}\)
Now we have:
Think about this logically: if you have a rectangular box, its volume equals length × width × height. In our case, we're dealing with a very thin layer, so we can think of it as:
\(\mathrm{Volume\,of\,layer} = \mathrm{Area\,of\,base} \times \mathrm{Thickness\,of\,layer}\)
This means that if we know the volume and the area, we can find the thickness by rearranging:
\(\mathrm{Thickness} = \mathrm{Volume} ÷ \mathrm{Area}\)
This makes intuitive sense - if you spread the same amount of material over a larger area, it gets thinner. If you spread it over a smaller area, it gets thicker.
Now we can substitute our values:
\(\mathrm{Thickness} = \mathrm{Volume} ÷ \mathrm{Area}\)
\(\mathrm{Thickness} = 25 \mathrm{cubic\,centimeters} ÷ 10,000 \mathrm{square\,centimeters}\)
\(\mathrm{Thickness} = \frac{25}{10,000} \mathrm{centimeters}\)
\(\mathrm{Thickness} = 0.0025 \mathrm{centimeters}\)
Let's verify this makes sense: we're spreading a small amount of material (25 cubic centimeters) over a very large area (1 square meter), so we should expect a very thin layer. 0.0025 centimeters is indeed very thin - about the thickness of a few sheets of paper.
Looking at our answer choices:
Our calculated answer of 0.0025 centimeters matches choice E exactly.
The thickness of the melt layer is 0.0025 centimeters.
Answer: E) 0.0025
Students often struggle to recognize that this is fundamentally a volume calculation problem. They may not realize that \(\mathrm{Volume} = \mathrm{Area} \times \mathrm{Thickness}\), which can be rearranged to find \(\mathrm{Thickness} = \mathrm{Volume} ÷ \mathrm{Area}\). Instead, they might attempt to use irrelevant formulas or try to multiply the given values rather than divide them.
Many students overlook that the volume is given in cubic centimeters while the area is given in square meters. They may proceed directly with the calculation without converting units, leading to an incorrect answer that doesn't match any of the given choices. This is a critical step that must be planned before starting calculations.
When converting 1 square meter to square centimeters, students commonly make the error of using 100 square centimeters instead of 10,000 square centimeters. They forget that area conversion requires squaring the linear conversion factor: \((100 \mathrm{cm})^2 = 10,000 \mathrm{cm}^2\), not just \(100 \mathrm{cm}^2\).
When calculating \(25 ÷ 10,000\), students may struggle with decimal placement or lose track of zeros. Common mistakes include getting 0.025 instead of 0.0025, or miscounting the number of decimal places in the final answer.
Students who made the unit conversion error (using 100 instead of 10,000) will calculate \(25 ÷ 100 = 0.25\) and select choice C, which appears reasonable but is incorrect due to the earlier conversion mistake. This demonstrates how errors compound through the solution process.