Question

The area of a telescope lens is \(7903 \mathrm{~mm}^{2}\). (a) What is the area in square feet \(\left(\mathrm{ft}^{2}\right) ?\) (b) If it takes a technician \(45 \mathrm{~s}\) to polish \(135 \mathrm{~mm}^{2},\) how long does it take her to polish the entire lens?

Short answer

The area in square feet is 0.08514 ft^2. Total polishing time is 2633.299 seconds.

Step-by-step solution

Convert Area to Square Feet

First, convert the area from square millimeters (\text{mm}^2) to square feet (\text{ft}^2). We use the conversion factor: 1 \text{mm}^2 = 1.07639 \times 10^{-5} \text{ft}^2. So, we multiply the area in \text{mm}^2 by this conversion factor:\[ 7903 \text{mm}^2 \times 1.07639 \times 10^{-5} \text{ft}^2/\text{mm}^2 = 0.08514 \text{ft}^2 \]

Determine Polishing Time per Square Millimeter

To find the polishing time for the entire lens, we need to determine how long it takes to polish one square millimeter. Given that it takes 45 seconds to polish 135 mm^2, we find the time per mm^2 as follows:\[ \text{Polishing time per mm}^2 = \frac{45 \text{s}}{135 \text{mm}^2} = 0.333 \text{s/mm}^2 \]

Calculate Total Polishing Time

Now, use the polishing time per mm^2 to find the total time for 7903 mm^2. Multiply the area by the time per mm^2:\[ 7903 \text{mm}^2 \times 0.333 \text{s/mm}^2 = 2633.299 \text{s} \]

Key concepts

Unit Conversion

Unit conversion is the process of converting one unit of measurement to another. It's a crucial skill in many fields, including science, engineering, and daily life. For this exercise, we converted the area of a telescope lens from square millimeters \( \mathrm{mm}^2 \) to square feet \( \mathrm{ft}^2 \). This involves using a conversion factor, a number that helps convert one unit to another. Here, we used 1 \( \mathrm{mm}^2 \) = 1.07639 \times 10^{-5} \( \mathrm{ft}^2 \).
To convert the area of the lens, multiply the given value by the conversion factor:
\[ 7903 \mathrm{mm}^2 \times 1.07639 \times 10^{-5} = 0.08514 \mathrm{ft}^2 \]This result shows that the lens area in square feet is 0.08514 \( \mathrm{ft}^2 \). Remember to always match the units correctly and double-check your conversion factor.

Area Calculation

Area calculation allows you to determine how much space a 2-dimensional surface occupies. For complex shapes or conversions, this might involve switching units or applying geometric formulas. In the lens area example, we converted from one unit to another without changing shape. When dealing with area, always ensure your units are consistent before calculating. In our exercise, knowing the area in \( \mathrm{mm}^2 \) and needing it in \( \mathrm{ft}^2 \), the conversion factor played a vital role. Remember, each side length squared results in squared units. So, converting linear measurements like mm to ft requires squaring the conversion factor too.

Time Calculation

Time calculation involves determining how long a task takes based on a rate and a total measure. Often, it's necessary to break a problem into steps. Here, we first figured out how long it takes to polish a single unit area, then extrapolated to the entire lens. We were given that 45 seconds is the time to polish 135 \( \mathrm{mm}^2 \). Dividing this, we get the time per unit area:
\[ \frac{45 \ \mathrm{s}}{135 \ \mathrm{mm}^2} = 0.333 \ \mathrm{s/mm}^2 \]
Next, to find the total time to polish the lens, multiply this rate by the total area:
\[ 7903 \ \mathrm{mm}^2 \times 0.333 \ \mathrm{s/mm}^2 = 2633.299 \ \mathrm{s} \]. This means it will take approximately 2633.299 seconds to polish the entire lens. Breaking problems into clear steps and using consistent units ensures accuracy.