Accepted answer
Answer included
Problem
A square sampling frame used in a biology lab (a quadrat) measures 180 cm by 180 cm. Find the area in m² and in cm².
Key idea
Converting lengths changes by the first power of the conversion factor, but converting areas changes by the square of the conversion factor. For example, since \(1\ \text{m} = 100\ \text{cm}\), then \(1\ \text{m}^2 = (100\ \text{cm})^2 = 10{,}000\ \text{cm}^2\).
Step-by-step solution
Step 1: Convert each side from centimeters to meters
Use \(1\ \text{m} = 100\ \text{cm}\).
\[
180\ \text{cm} \times \frac{1\ \text{m}}{100\ \text{cm}} = 1.80\ \text{m}
\]
Step 2: Compute area in square meters
For a square, area equals side times side.
\[
A = (1.80\ \text{m}) \cdot (1.80\ \text{m}) = 3.24\ \text{m}^2
\]
Step 3: Compute area in square centimeters
Using the original dimensions directly:
\[
A = (180\ \text{cm}) \cdot (180\ \text{cm}) = 32{,}400\ \text{cm}^2
\]
This matches the area-conversion rule:
\[
3.24\ \text{m}^2 \times 10{,}000\ \frac{\text{cm}^2}{\text{m}^2} = 32{,}400\ \text{cm}^2
\]
Summary table
| Quantity | Given | Conversion | Result |
|---|---|---|---|
| Side length | 180 cm | \(\times \frac{1\ \text{m}}{100\ \text{cm}}\) | 1.80 m |
| Area (in m²) | \((1.80\ \text{m}) \cdot (1.80\ \text{m})\) | multiply lengths | 3.24 m² |
| Area (in cm²) | \((180\ \text{cm}) \cdot (180\ \text{cm})\) | multiply lengths | 32,400 cm² |
Visualization
Final answers
- Area: \(3.24\ \text{m}^2\)
- Area: \(32{,}400\ \text{cm}^2\)
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