Question

A small hole in the wing of a space shuttle requires a \(20.7-\mathrm{cm}^{2}\) patch. (a) What is the patch's area in square kilometers ( \(\mathrm{km}^{2}\) )? (b) If the patching material costs NASA \(\$ 3.25 / \mathrm{in}^{2},\) what is the cost of the patch?

Short answer

The patch's area is \(2.07 \times 10^{-9} \text{ km}^2\), and the cost is \$10.42.

Step-by-step solution

- Convert \text{cm}^2 to m^2

First, convert the patch's area from square centimeters to square meters. We know that 1 square meter is equal to \text{10,000} square centimeters. Thus, we divide the patch's area by 10,000:\(\frac{20.7 \text{ cm}^2}{10,000} = 0.00207 \text{ m}^2\).

- Convert m^2 to km^2

Now, convert the area from square meters to square kilometers. We know that 1 square kilometer is equal to \text{1,000,000} square meters. Thus, we divide the area in square meters by 1,000,000:\(\frac{0.00207 \text{ m}^2}{1,000,000} = 2.07 \times 10^{-9} \text{ km}^2\).

- Convert cm^2 to in^2

Convert the patch's area from square centimeters to square inches. We need the factor for conversion, with 1 inch equal to 2.54 centimeters, therefore 1 square inch is equal to (2.54)^2 square centimeters. Thus, we divide the patch's area by (2.54)^2:\(\frac{20.7 \text{ cm}^2}{2.54^2} \approx 3.207 \text{ in}^2\).

- Calculate the Cost of the Patch

Finally, calculate the cost of the patch given that the patching material costs \$3.25 per square inch. Multiply the patch area in square inches by the cost per square inch:\(3.207 \text{ in}^2 \times \$3.25/\text{ in}^2 = \$10.42\).

Key concepts

Area Conversion

Area conversion is an essential skill in science and engineering, especially when dealing with different units of measurement. It involves converting the area from one unit to another while maintaining the same scale. In our example, the initial patch area was given in square centimeters (\text{cm}^2). To convert this area to square kilometers (\text{km}^2), we performed the following steps:

• First, we converted \text{cm}^2 to square meters (\text{m}^2). Since 1 square meter is equivalent to 10,000 square centimeters, we divided the initial area by 10,000:

\[ \frac{20.7 \text{ cm}^2}{10,000} = 0.00207 \text{ m}^2 \]

• Next, we converted \text{m}^2 to \text{km}^2. One square kilometer is equal to 1,000,000 square meters. So, we divided the area in \text{m}^2 by 1,000,000:

\[ \frac{0.00207 \text{ m}^2}{1,000,000} = 2.07 \times 10^{-9} \text{ km}^2 \]
This step-by-step process ensures that you're converting units accurately without losing any information.

Metric System

Understanding the metric system is fundamental for solving many scientific problems. The metric system is an international system of measurement that uses units like meters, liters, and grams. It's based on powers of ten, making it easy to convert between units by shifting the decimal point.

• The basic unit for length is the meter (m), and for area, we use square meters (\text{m}^2).
• To convert square meters to square kilometers (\text{km}^2), remember that 1 km = 1,000 m, so: \[ 1 \text{ km}^2 = (1,000 \text{ m})^2 = 1,000,000 \text{ m}^2 \]

In the example, we had to convert from \text{cm}^2 to \text{km}^2 by first converting to \text{m}^2 and then to \text{km}^2. Understanding these conversions in the metric system helps you manage complex problems intuitively.

Cost Calculation

Cost calculation involves determining the expenses based on the given cost parameters. In our exercise, NASA's patching material cost is provided per square inch. To find the total cost, follow these steps:

• First, convert the patch's area from square centimeters to square inches. We know that 1 inch equals 2.54 centimeters, thus 1 square inch equals \[ (2.54)^2 \text{ cm}^2. \]
• Using this, convert the given area: \[ \frac{20.7 \text{ cm}^2}{2.54^2} \thickapprox 3.207 \text{ in}^2 \]

Finally, multiply the area in square inches by the cost per square inch to get the total cost: \[ 3.207 \text{ in}^2 \times 3.25 \text{ dollars/in}^2 = 10.42 \text{ dollars} \]
Through these calculations, you can efficiently determine the expenses required for specific tasks using accurate unit conversions and basic arithmetic.