Question

1. what is the length of this screw in cm using the proper number of significant digits?

a. 5.1 cm
b. 5 cm
c. 5.111 cm
d. 5.10 cm

Explanation:

Step1: Identify the scale

The ruler is marked in centimeters, with each centimeter divided into 10 smaller divisions (millimeters). So the least count is 0.1 cm.

Step2: Read the initial and final positions

The screw starts at the 0 cm mark (or the start of the visible scale near 0, considering the alignment) and ends at the 5.1 cm mark (since the tip is at the 5.1 cm position when considering the scale divisions). We need to consider significant digits. The measurement should reflect the precision of the instrument. The ruler can measure to the tenths place (0.1 cm), and the estimated digit (if any) - but here, the reading is 5.1 cm which has two significant digits? Wait, no - wait, the options: let's check the scale again. Wait, the screw's top is at the 0 cm (or the 0.0 cm) and the bottom at 5.1 cm? Wait, looking at the image, the screw's head is at around the 0.0 cm mark (the top dashed line) and the tip at the 5.1 cm mark (the bottom dashed line). Now, significant digits: when measuring, the number of significant digits includes all certain digits plus one estimated digit. The ruler has divisions of 0.1 cm (each small line is 0.1 cm). So the certain digit is 5 cm, and the next digit is estimated as 0.1 cm? Wait, no, the tip is at the 5.1 cm mark? Wait, the options: a is 5.1 cm, b is 5 cm, c is 5.111 cm, d is 5.10 cm. Wait, the ruler's precision is 0.1 cm (since each cm is divided into 10 mm, so 0.1 cm per division). So when measuring, we can record to the tenths place, and if the object ends exactly on a division, we can add a zero as a placeholder for significant digits? Wait, no. Wait, the screw's length: from the top (0.0 cm) to the bottom (5.1 cm)? Wait, no, looking at the ruler, the numbers are 1,2,3,4,5,6 cm. Wait, maybe the top of the screw is at the 0.0 cm (the first dashed line) and the bottom at 5.1 cm (the second dashed line). Wait, the scale: each cm is divided into 10 mm, so each small line is 1 mm = 0.1 cm. So the reading is 5.1 cm? But wait, the options: d is 5.10 cm. Wait, maybe the ruler is precise to 0.01 cm? No, standard rulers have 1 mm divisions, so 0.1 cm precision. But when reporting, if the measurement is exactly on a division, we can write it as 5.10 cm to show that the hundredths place is estimated as 0? Wait, no. Wait, significant digits: the number of significant digits is determined by the precision. Let's think again. The screw's length: from the start (let's say 0.00 cm) to the end (5.10 cm)? Wait, maybe the image shows that the tip is at the 5.10 cm mark? Wait, the options: d is 5.10 cm, a is 5.1 cm. Wait, maybe the correct answer is a? Wait, no, let's check the scale. Wait, the ruler has cm marks (1,2,3,4,5,6) and between each cm, 10 mm. So the screw's head is at the 0 cm (the top dashed line) and the tip at the 5.1 cm (the bottom dashed line). But when measuring, the first digit is 5 cm (certain), the next digit is 0.1 cm (certain, because it's on the 1 mm mark). Wait, no, if it's exactly on the 5.1 cm mark, then the measurement is 5.1 cm, but if we consider significant digits, maybe 5.10 cm? Wait, no, the precision is 0.1 cm, so the number of significant digits: 5.1 has two significant digits? No, 5.1 has two? Wait, 5 is certain, 1 is estimated? Wait, no, if the ruler has 1 mm divisions, then we can measure to the mm, so 5.1 cm (51 mm) is precise. But the options: d is 5.10 cm. Wait, maybe the question is about significant digits, and when the measurement is made with a ruler that has 1 mm divisions, the length is 5.10 cm? Wait, no, maybe I'm overcomplicating. Let's look at the options. The screw's length: from the top (0 cm) to the bottom (5.1 cm? Or 5.10 cm?). Wait, the image: the top dashed line is at the 0 cm (or the start of the scale, maybe 0.0 cm) and the bottom dashed line is at 5.1 cm? Wait, no, the ruler's scale: the numbers are 1,2,3,4,5,6 cm. Wait, maybe the top of the screw is at the 0.0 cm (the first dashed line) and the bottom at 5.10 cm (the second dashed line, which is at the 5.1 cm mark but with a zero for significant digits). Wait, the options: d is 5.10 cm, a is 5.1 cm. Wait, maybe the correct answer is a? No, wait, let's check the significant digits. The ruler can measure to the tenths place (0.1 cm), so the measurement is 5.1 cm (two significant digits), but if we consider that the zero after the decimal is a placeholder, maybe 5.10 cm (three significant digits). Wait, the question says "proper number of significant digits". Let's think: when using a ruler with 1 mm divisions, the length is measured as 5.1 cm (if the tip is at the 5.1 cm mark) or 5.10 cm? Wait, no, 5.1 cm has two significant digits, 5.10 has three. The ruler's precision is 0.1 cm, so the uncertain digit is in the hundredths place. So when measuring, we can record 5.10 cm to show that we estimated the hundredths place as 0. So the correct answer is d? Wait, no, maybe the image shows that the tip is at the 5.1 cm mark, so 5.1 cm. But the options: a is 5.1 cm, d is 5.10 cm. Wait, maybe the answer is a. Wait, let's re-express:

The screw's length is from the initial position (let's say 0.0 cm) to the final position (5.1 cm). So the length is 5.1 - 0.0 = 5.1 cm. Now, significant digits: 5.1 has two significant digits? No, 5 and 1 are significant, so two? Wait, no, 5.1 has two significant digits? Wait, 5 is a significant digit, 1 is a significant digit, so two. But option d is 5.10, which has three. Wait, maybe the ruler is more precise. Wait, the image: the ruler has cm marks and between each cm, 10 mm, so each mm is 0.1 cm. So the screw's tip is at the 5.1 cm mark (the 5 cm + 1 mm mark). So the length is 5.1 cm. But if we consider that the zero after the decimal is a placeholder for significant digits (indicating that the measurement is precise to the hundredths place), but the ruler can't measure to the hundredths place. Wait, maybe the question is designed to have the answer as a: 5.1 cm.

Wait, maybe I made a mistake. Let's check the options again. The question is "What is the length of this screw in cm using the proper number of significant digits?"

Option a: 5.1 cm (two significant digits)

Option b: 5 cm (one significant digit)

Option c: 5.111 cm (four significant digits, too precise)

Option d: 5.10 cm (three significant digits)

The ruler has a precision of 0.1 cm (1 mm), so the measurement should be reported to the tenths place, and the number of significant digits should reflect that. So 5.1 cm (two significant digits) is appropriate, because the 5 is certain, the 1 is estimated (but in this case, it's on the mark, so maybe 5.10? Wait, no, if it's exactly on the 5.1 cm mark, then the hundredths place is 0, so 5.10 cm (three significant digits) because the zero is a significant digit indicating precision. Wait, the rule is: trailing zeros after the decimal are significant. So 5.10 has three significant digits, 5.1 has two. The ruler can measure to 0.1 cm, so the uncertainty is in the hundredths place. So when we measure, we can write 5.10 cm to show that we know the hundredths place is 0 (even though the ruler can't measure it, but maybe the question considers that). Wait, maybe the correct answer is d: 5.10 cm. But I'm confused. Wait, let's look at the image again. The screw's tip is at the 5.1 cm mark (the line at 5.1 cm). So the length is 5.1 cm. But if we consider significant digits, when the measurement is made with a ruler that has 1 mm divisions, the length is 5.1 cm (two significant digits) or 5.10 cm (three)? Wait, the correct approach is: the number of significant digits is determined by the precision of the instrument. The ruler has a least count of 0.1 cm, so we can measure to the tenths place, and the estimated digit is in the hundredths place. So if the object ends exactly on a division, we can write it as 5.10 cm to indicate that the hundredths place is 0 (estimated). So the proper number of significant digits would be three, so 5.10 cm. So the answer is d? Wait, no, maybe the question is simpler. Let's see: the screw starts at 0.0 cm and ends at 5.1 cm, so length is 5.1 cm. But the options: a is 5.1, d is 5.10. Maybe the answer is a. I think I need to go with the most appropriate. Let's check the scale: each cm is divided into 10 mm, so 0.1 cm per division. So the reading is 5.1 cm (5 cm + 1 mm). So the length is 5.1 cm, which has two significant digits. So the answer is a.

Answer:

a. 5.1 cm