Q.Calculate the solid angle subtended by the periphery of an area of 1 cm$^2$ at a point situated symmetrically at a distance of 5 cm from the area.

The concept here is significant figures (or significant digits), which tell us how precise a measurement is. The rules are simple but often misapplied:

• All non-zero digits are significant.
• Zeros between non-zero digits are significant.
• Leading zeros (to the left of the first non-zero digit) are not significant — they’re just placeholders.
• Trailing zeros after a decimal point are significant because they indicate the precision of the measurement.

In 0.020260, the zeros at the very beginning are just placeholders, but the zeros inside and at the end matter.

Let’s count step by step:

1. Identify the first non-zero digit.

   The number is 0.020260. The first non-zero digit is the 2 after the decimal (the hundredths place). All zeros before it (the leading 0 before the decimal and the first 0 after the decimal) are not significant.

2. Count from that first non-zero digit onward.

   Starting from the 2:

   • 2 (significant)
   • 0 (between 2 and 2, so significant)
   • 2 (significant)
   • 6 (significant)
   • 0 (trailing zero after the decimal, so significant) …

The problem is about significant figures in a calculation. When you multiply or divide numbers, the result cannot be more precise than the least precise measurement involved. That means the number of significant figures in the final answer is determined by the term with the fewest significant figures.

Let’s identify the significant figures in each number:

• $0.004560$ has 4 significant figures (the leading zeros don’t count, but the trailing zero after the decimal does).
• $1200$ has 4 significant figures (no decimal point, so trailing zeros are ambiguous — but here it’s written without a decimal, so conventionally it has 4 significant figures; however, in many exam contexts, $1200$ is taken as having 2 significant figures unless specified. Let’s check carefully: the problem gives $1200$ without a decimal, so the safest interpretation is that it has 2 significant figures, because the trailing zeros are not significant. But wait — in the given expression, $1200$ appears alongside $0.004560$ (4 sig figs) and $3.00 \times 10^{-2}$ (3 sig figs). The most common convention in such problems is that $1200$ has 2 significant figures. We’ll verify by looking at the options — they all have 2, 3, or 4 significant figures, so we must decide.)
• $3.00 \times 10^{-2}$ has 3 significant figures (the zeros after the decimal are significant).

Now, the term with the fewest significant figures is $1200$ with 2 significant figures (if we take the standard convention). So the final answer must be rounded to 2 significant figures. …

When we subtract two measured quantities, the result cannot be more precise than the least precise measurement. This is a fundamental rule of significant figures: the number of decimal places in the answer is limited by the measurement with the fewest decimal places. Here, 43.4 m has one decimal place (the tenths place is certain), while 3.532 m has three decimal places. So the difference must be reported to one decimal place only.

Let’s work through it step by step:

1. Perform the subtraction exactly

This is the raw arithmetic result, but it implies a precision we don’t actually have.

2. Identify the limiting precision

   The rod’s original length, 43.4 m, is given to the nearest tenth of a metre. That means the true length lies between 43.35 m and 43.45 m. The piece cut, 3.532 m, is given to the nearest thousandth, so its true length lies between 3.5315 m and 3.5325 m.

3. Consider the range of possible remainders

   The smallest possible remainder occurs when the original rod is at its shortest and the cut piece is at its longest:

The largest possible remainder occurs when the original rod is at its longest and the cut piece is at its shortest:

So the true remainder lies somewhere between 39.8175 m and 39.9185 m.

4. Round to the correct number of decimal places …

We are given a rod of length 43.4 m and a piece of length 3.532 m is cut from it. The question asks for the length of the remaining rod, reported with the correct number of significant figures. This is not just a simple subtraction — we must respect the precision of the measurements.

Concept and Intuition:

When adding or subtracting measurements, the result cannot be more precise than the least precise measurement. Here, 43.4 m has one decimal place (tenths), while 3.532 m has three decimal places (thousandths). The subtraction should therefore be rounded to the tenths place. Many students mistakenly keep all digits from the calculation, but that would imply a false precision.

Step-by-step solution:

1. Perform the subtraction exactly:

This is the raw arithmetic result.

2. Identify the least precise measurement:

   • 43.4 m is measured to the nearest tenth (one decimal place).
   • 3.532 m is measured to the nearest thousandth (three decimal places). The least precise is 43.4 m, with uncertainty in the tenths place.

3. Apply the rule for subtraction:

   The result must be rounded to the same decimal place as the least precise measurement — the tenths place. Look at the digit in the hundredths place of 39.868, which is 6. Since 6 ≥ 5, we round up the tenths digit from 8 to 9.

4. Round the result: …

Concept & Intuition

The key idea is significant figures in division. When you multiply or divide measurements, the result cannot be more precise than the least precise measurement. The volume (2.43 cm³) has only 3 significant figures, so the density must be rounded to 3 significant figures — even if the raw calculation gives more digits. Many students forget this and just report the full calculator display, which leads to a wrong choice.

Step-by-step solution

1. Write the formula for density Density $\rho$ is mass $m$ divided by volume $V$:

2. Plug in the given values $m = 5.318 \, \text{g}$ (4 significant figures) $V = 2.43 \, \text{cm}^3$ (3 significant figures)

3. Perform the division Using a calculator:

4. Apply the significant figure rule The volume has only 3 significant figures, so the density must be rounded to 3 significant figures. Look at the fourth digit: 2.188… — the third digit is 8, and the next digit is 8 ≥ 5, so we round up: $2.188 \rightarrow 2.19$ …

When we perform calculations with measured quantities, it's crucial to express the result with an appropriate level of precision. This is where the concept of significant figures comes in. Significant figures indicate the reliability of a measurement. If a measurement is given as $11.2$ cm, it implies that the measurement is reliable up to the first decimal place, and there's some uncertainty in the second decimal place and beyond.

When adding or subtracting measurements, the rule for significant figures is based on the number of decimal places. The idea is that the sum or difference cannot be more precise than the least precise measurement involved in the calculation. If one measurement is known only to the tenths place, the sum cannot be reliably known to the hundredths or thousandths place.

The specific rule is:

The result of an addition or subtraction should be rounded to the same number of decimal places as the measurement with the fewest decimal places.

Let's apply this to the given problem.

1. Identify the given lengths:

   We are given two rod lengths:

   • Length of the first rod, $L_1 = 13.244$ cm
   • Length of the second rod, $L_2 = 11.2$ cm

2. Perform the addition:

   To find the combined length, we add the two lengths:

   $L_{combined} = L_1 + L_2 = 13.244 \text{ cm} + 11.2 \text{ cm} = 24.444 \text{ cm}$

3. Determine the number of decimal places in each original measurement:

   • $L_1 = 13.244$ cm has three digits after the decimal point (2, 4, 4). So, it has 3 decimal places.
   • $L_2 = 11.2$ cm has one digit after the decimal point (2). So, it has 1 decimal place.

4. Apply the rule for significant figures in addition:

   According to the rule, the result must be rounded to the same number of decimal places as the measurement with the fewest decimal places. Comparing 3 decimal places and 1 decimal place, the fewest is 1 decimal place.

   Therefore, our calculated sum $24.444$ cm must be rounded to 1 decimal place. …

The key concept is significant figures (also called significant digits). These are the digits in a number that carry meaningful information about its precision. When a number is written in scientific notation — like $3.78 \times 10^{22}$ — the exponent ($10^{22}$) only tells us the order of magnitude (how large or small the number is). It does not affect how many digits are considered reliable or measured. All the significant figures are in the coefficient (the number before the $\times 10$).

A common mistake is to think the exponent contributes to the count of significant figures. For example, someone might see $10^{22}$ and think “22” is part of the precision, but that’s wrong — the exponent is just a placeholder for the decimal point.

Let’s work through it:

1. Identify the coefficient.

   The number is $3.78 \times 10^{22}$. The coefficient is $3.78$.

2. Count the digits in the coefficient.

   $3.78$ has three digits: 3, 7, and 8. All are non-zero, so each is significant.

3. Ignore the exponent.

   The $10^{22}$ part tells us the number is $378\,000\,000\,000\,000\,000\,000\,000$ (378 followed by 20 zeros), but those zeros are not measured — they are just placeholders. The precision is still only three digits.

4. Select the matching option. …

The key idea here is the rule for significant figures in addition and subtraction: the result cannot have more decimal places than the measurement with the fewest decimal places. This is different from multiplication/division, where you count total significant figures. Many students mistakenly apply the multiplication rule to addition, which is the classic pitfall.

Let’s work through it step by step.

1. Identify the decimal places of each term.

   • 12.0 has 1 decimal place (the tenths place).
   • 19.034 has 3 decimal places (thousandths).
   • 2.0143 has 4 decimal places (ten-thousandths). The least precise measurement is 12.0, with only 1 decimal place.

2. Perform the addition exactly.

This is the raw sum.

3. Apply the addition rule. Since the least precise number has 1 decimal place, the sum must be rounded to 1 decimal place.

(The digit after the tenths place is 4, so we round down.)

4. Count the significant figures in the rounded result. …

The concept here is significant figures — the digits in a number that carry meaningful information about its precision. The rules for counting them are straightforward, but the trickiest part is handling zeros. Zeros can be significant or not, depending on where they appear.

For $2.0400$, let’s apply the rules step by step.

1. All non-zero digits are significant.

   The digit 2 is non-zero, so it counts. That’s 1 significant figure so far.

2. Zeros between non-zero digits are significant.

   Here, the zero between 2 and 4 (the first zero after the decimal) is between two non-zero digits? Actually, check: the digits are 2, ., 0, 4, 0, 0. The zero immediately after the decimal is between 2 and 4? No — 2 is before the decimal, 0 is after. But the rule applies to zeros between non-zero digits within the number. Since 2 and 4 are separated only by a decimal point and a zero, that zero is indeed between them, so it is significant. That’s 2 significant figures now.

3. Trailing zeros after a decimal point are significant.

   Look at the end: ...400. The last two zeros come after the decimal and after the non-zero digit 4. These are trailing zeros in a decimal number, so they are significant. That adds 2 more significant figures (the two zeros at the end).

4. Count them all:

   • 2 → 1
   • 0 (between 2 and 4) → 1
   • 4 → 1
   • 0 (trailing) → 1
   • 0 (trailing) → 1 …

The key idea behind significant figures is that they count only the digits that carry meaningful information about precision. For a large number like 234555359, the first three significant figures are the digits 2, 3, and 4 — because the number starts with a non-zero digit. The rest of the digits are placeholders that get replaced by zeros after rounding.

When rounding to a certain number of significant figures, you look at the digit immediately after the last one you want to keep. If that digit is 5 or greater, you round up the last kept digit; otherwise, you leave it as is.

1. Identify the first three significant figures.

   The number is 234555359. The first digit (2) is the most significant, followed by 3, then 4. So the first three significant figures are 2, 3, and 4.

2. Locate the fourth digit (the one that decides rounding).

   The fourth digit is the next one after the third significant figure. Here, after 2-3-4 comes the digit 5 (the fourth digit). So we look at this 5.

3. Apply the rounding rule.

   Since the fourth digit is 5 (which is ≥ 5), we round up the third significant figure (which is 4) by 1. That makes it 5. All digits after the third significant figure become zeros.

4. Write the rounded number. …