Question

use the rules for working with measurements to give your answer to the appropriate accuracy and/or precision. find the total amount (area) of paper used for labels for 1,000 cans like the one shown in the figure below (in cm²). 2.27 cm 7.48 cm

Explanation:

Step1: Identify the shape of the label

The label on a can is a rectangle when unrolled, but its area is equivalent to the lateral (curved) surface area of a cylinder. The formula for the lateral surface area of a cylinder is \( A = 2\pi rh \), where \( r \) is the radius and \( h \) is the height. Here, the radius \( r = 2.27 \, \text{cm} \) and the height \( h = 7.48 \, \text{cm} \).

Step2: Calculate the lateral surface area for one can

Substitute \( r = 2.27 \) and \( h = 7.48 \) into the formula:
\( A = 2 \times \pi \times 2.27 \times 7.48 \)
First, calculate \( 2 \times 2.27 = 4.54 \). Then, \( 4.54 \times 7.48 \approx 33.9592 \). Multiply by \( \pi \): \( 33.9592 \times \pi \approx 33.9592 \times 3.1416 \approx 106.63 \, \text{cm}^2 \) (for one can).

Step3: Calculate the total area for 1,000 cans

Multiply the area of one can by 1,000:
\( \text{Total Area} = 106.63 \times 1000 = 106630 \, \text{cm}^2 \) (we check significant figures: the given values \( 2.27 \) (3 sig figs) and \( 7.48 \) (3 sig figs) suggest the answer should have 3 sig figs, but let's verify the calculation more precisely).

Wait, let's recalculate the lateral surface area more accurately:
\( 2 \times \pi \times 2.27 \times 7.48 = 2 \times 7.48 \times 2.27 \times \pi \)
\( 2 \times 7.48 = 14.96 \)
\( 14.96 \times 2.27 = 14.96 \times 2 + 14.96 \times 0.27 = 29.92 + 4.0392 = 33.9592 \)
\( 33.9592 \times \pi \approx 33.9592 \times 3.14159265 \approx 106.631 \, \text{cm}^2 \) per can.

For 1,000 cans: \( 106.631 \times 1000 = 106631 \, \text{cm}^2 \). Rounding to appropriate precision (since the measurements have three significant figures), the total area is approximately \( 1.07 \times 10^5 \) or \( 107000 \)? Wait, no—let's check the multiplication again. Wait, \( 2.27 \) is radius, \( 7.48 \) is height. Let's confirm the formula: lateral surface area of cylinder is \( 2\pi rh \), which is correct.

Wait, maybe I made a mistake in the first calculation. Let's do it step by step with more precision:

\( 2 \times \pi = 6.283185307 \)
\( 6.283185307 \times 2.27 = 6.283185307 \times 2 + 6.283185307 \times 0.27 = 12.56637061 + 1.696460033 = 14.26283064 \)
\( 14.26283064 \times 7.48 = 14.26283064 \times 7 + 14.26283064 \times 0.48 = 99.83981448 + 6.846158707 = 106.6859732 \, \text{cm}^2 \) per can.

Then, for 1,000 cans: \( 106.6859732 \times 1000 = 106685.9732 \, \text{cm}^2 \). Now, considering significant figures: \( 2.27 \) (3 sig figs) and \( 7.48 \) (3 sig figs), so the result should have 3 sig figs. \( 106685.9732 \) rounded to 3 sig figs is \( 1.07 \times 10^5 \) or \( 107000 \)? Wait, \( 106686 \) rounded to 3 significant figures: the first three significant figures are 1, 0, 6? Wait no, 106686: the first non-zero digit is 1, then 0, 6, 6... Wait, no, 106686 is \( 1.06686 \times 10^5 \). To three significant figures, it's \( 1.07 \times 10^5 \) (since the fourth digit is 6, which rounds up the third digit from 6 to 7). So \( 1.07 \times 10^5 = 107000 \, \text{cm}^2 \). Wait, but maybe the problem expects us to use the given numbers as is, without worrying about significant figures, just multiply. Let's recalculate:

\( 2 \times \pi \times 2.27 \times 7.48 \times 1000 \)

First, multiply all the numbers: \( 2 \times 2.27 \times 7.48 \times 1000 \times \pi \)

\( 2 \times 2.27 = 4.54 \)
\( 4.54 \times 7.48 = 33.9592 \)
\( 33.9592 \times 1000 = 33959.2 \)
\( 33959.2 \times \pi \approx 33959.2 \times 3.14159265 \approx 106685.97 \, \text{cm}^2 \)

So the total area is approximately \( 106686 \, \text{cm}^2 \), which can be rounded to \( 107000 \) (3 sig figs) or kept as \( 106686 \). But let's check the problem statement: it says "use the rules for working with measurements to give your answer to the appropriate accuracy and/or precision". The measurements \( 2.27 \) (three significant figures) and \( 7.48 \) (three significant figures) suggest that the answer should have three significant figures. So \( 106686 \) rounded to three significant figures is \( 1.07 \times 10^5 \) or \( 107000 \). Wait, \( 106686 \) is \( 1.06686 \times 10^5 \), so rounding to three significant figures: look at the fourth digit, which is 6, so we round up the third digit (6) to 7, giving \( 1.07 \times 10^5 \) or \( 107000 \, \text{cm}^2 \).

Answer:

\( 107000 \) (or more precisely \( 106686 \), but with three significant figures, \( 1.07 \times 10^5 \) or \( 107000 \))