Photographic film is manufactured by applying a very thin layer of a viscous solution, called a melt, to a transparent,...

1. Translate the problem requirements: We need to find the thickness of a layer when 25 cubic centimeters of material is spread evenly over 1 square meter of surface area. This is essentially asking: if I have a certain volume spread over a known area, how thick is the resulting layer?
2. Convert units to be consistent: Since the answer choices are in centimeters, convert the area from square meters to square centimeters to match the volume units.
3. Apply the volume relationship: Use the fundamental relationship that \(\mathrm{Volume} = \mathrm{Area} \times \mathrm{Thickness}\), which means \(\mathrm{Thickness} = \mathrm{Volume} ÷ \mathrm{Area}\).
4. Calculate and verify with answer choices: Perform the division and check that the result matches one of the given options.

Execution of Strategic Approach

1. Translate the problem requirements

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:

• Volume of melt = 25 cubic centimeters
• Area of support material = 1 square meter
• We need to find: thickness of the layer in centimeters

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

2. Convert units to be consistent

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:

• Volume = 25 cubic centimeters
• Area = 10,000 square centimeters
• Need to find: thickness in centimeters

3. Apply the volume relationship

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.

4. Calculate and verify with answer choices

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:

1. 25
2. 2.5
3. 0.25
4. 0.025
5. 0.0025

Our calculated answer of 0.0025 centimeters matches choice E exactly.

Final Answer

The thickness of the melt layer is 0.0025 centimeters.

Answer: E) 0.0025

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the basic volume-area-thickness relationship

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.

2. Failing to identify the unit conversion requirement

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.

Errors while executing the approach

1. Incorrect unit conversion 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 \(100 \mathrm{cm}^2\).

2. Arithmetic errors in division

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.

Errors while selecting the answer

1. Choosing an answer that results from incomplete unit conversion

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.