# Section 3C

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## Rounding ( Brief review on p.152)

Step 1 Decide about the last place to be kept.

Step 2 If the $$next$$ digit is below 5, round down, while if the next digit is 5 and above, round up.

Example: 398.3 rounded to the nearest one is 398

Example: 398.3 rounded to the nearest ten is 400

Exercise 15 p.160 Round to the nearest whole number

$$3.45$$ $$\approx 3$$

$$87.737$$ $$\approx 88$$

$$0.12$$ $$\approx 0$$

$$184.73$$ $$\approx 185$$

$$1945.1$$ $$\approx 1945$$

$$2.5001$$ $$\approx 3$$

$$6.495$$ $$\approx 6$$

$$1499.5$$ $$\approx 1500$$

$$-13.6$$ $$\approx -14$$

### Counting Significant Digits

Exercise 18,19,20,21,22,23,24 p.160 State the number of significant digits

"90 miles per hour" 1 significant digits

"90.001 miles per hour" 5 significant digits

"0.01 centimeter" 1 significant digit

"3.32234 miles" 6 significant digits

"450,000 years" 2 significant digits

"$$1.2 \times 10^4$$ seconds" 2 significant digits

"0.000203 meter" 3 significant digits

## Rounding with Significant Digits

Exercise 29 p.160

Multiply $$45 \times 32.1$$, and state the answer with 3 significant digits.

Solution. $$45 \times 32.1$$ $$= 1444.5$$ $$\approx 1440$$<---- your answer
We have rounded 1444.5 so that there are only 3 significant digits. We may, and in this case we have to, declare the zero in the right non-significant.

## Rounding with Significant Digits

Exercise 31 p.160

Divide $$231.89 \div 0.034$$, and state the answer with 2 significant digits.

Solution.
$$231.89 \div 0.034$$ $$= 6820.29411765$$ $$\approx 6800$$<---- your answer
We have rounded 6820.29411765 so that there are only 2 significant digits. We may, and in this case we have to, declare the zeros in the right non-significant.

## Absolute and Relative Errors

Exercise 47 p.160

Your true height is 68.0 inches (5'8"), but a nurse in a doctor's office measures your height as 67.5 inches. Find absolute and relative errors.

$$\text{absolute error} = \text{measured value} - \text{true value}$$
$$= 67.5 - 68.0$$ $$= -0.5$$ inches <--- your answer

$$\text{relative error} = \frac{\text{absolute error}}{\text{true value}} \times 100 \%$$ $$= \frac{-0.5}{68} \times 100 \%$$ $$= -0.7\%$$ <--- your answer

### Absolute and Relative Errors

Exercise 50 p.160

An order of fish and chips is supposed to cost $$8.95$$, but the server accidentally keys in a price of $$9.95$$. Find absolute and relative errors.

$$\text{absolute error} = \text{measured value} - \text{true value}$$
$$= 9.95 - 8.95$$ $$= 1$$ dollar <--- your answer

$$\text{relative error} = \frac{\text{absolute error}}{\text{true value}} \times 100 \%$$ $$= \frac{1}{8.95} \times 100 \%$$ $$= 11.2\%$$ <--- your answer

Subtlety. Here $$1 = 1.00$$ with significant zeros, therefore $$3$$ digits in the answer

## Accuracy and Precision

Exercise 55 p.161

Your true height is $$70.50$$ inches. A tape measure that can be read to the nearest $$\frac{1}{8}$$ inch gives your height as $$70\frac{3}{8}$$ inches. A new laser devise at a doctor's office which gives readings to the nearest $$0.05$$ inch gives your height as $$70.90$$ inches. Which one is more accurate, and which one is more precise.

##### Exercise 55 p.161 ... Solution

The laser devise is more precise, because it provides more detail since
$$0.05$$ is smaller that $$\frac{1}{8}$$.

In order to find out which measurement is more accurate, we need to compare absolute errors.

With the tape measure, absolute error is $$70\frac{3}{8} - 70.50 = -\frac{1}{8}$$.

In doctor's office, absolute error is $$70.90-70.50 = 0.4$$

Since $$0.4$$ is bigger than $$\frac{1}{8}$$, the tape measure is more accurate.

## Combining numbers

Exercise 63 p.161

As you drive down the freeway, a sign tells you that it is 36 miles to city hall. Your destination lies 2.2 miles beyond city hall. How much further do you have to drive?

Solution. Clearly, 36+2.2=38.2. However, 36 is precise to 1 mile, while 2.2 is precise up to 0.1 mile. We thus have to round 38.2 to the nearest one: $$38.2 \approx 38$$ <---- your answer.

## Combining numbers

Exercise 65 p.161

What is the per capita cost of 2.1 million recreation center in a city with 120,345 people?

Solution.
Clearly, $$(2.1 \times 10^6 ) \div 120345 =17.4498317338$$. However, 2.1 million has 2 significant digits. Despite of 120,345 has 6 significant digits, we have to stick to only 2. We thus have to round 17.4498317338 to the nearest one: $$17.4498317338 \approx 17$$ <---- your answer.

## Propagation of Error

Exercise 75 p.161 (modified)

Suppose you want to cut 30 identical boards of size 8 feet. The procedure is to measure and cut first board, then use the first board to measure and cut the second board, then use the second board to measure and cut the third board, and so on.
a. What is the possible lengths of the 30th board if each time you cut a board there is a maximum error of $$\pm 1$$ inch?
b. What is the possible lengths of the 30th board if each time you cut a board there is a maximum error of $$\pm\ 1\%$$?

##### Solution for a.

The maximum error of $$\pm 1$$ inch may compound.

Thus, in the worst case scenario, the 30th board is

$$30 \times 1 = 30$$ inches shorter or longer than required.

Since 8 feet = 96 inches, the 30th board may be as short as $$96 -30 =66$$ inches,

and as long as $$96 +30 =126$$ inches long.

Answer: Between 66 and 126 inches.

##### Solution for b.

The maximum error of $$\pm 1 \%$$ may compound.

That may make every next board 1.01 times longer than the previous one.

The 30th board in this way becomes $$(1.01)^{30}$$ times longer because we multiply the length every time by a factor of 1.01 (or $$0.99$$ ), and we do that 30 times in a row.

$$1.01^{30}=1.34784891533$$, and $$0.99^{30}= 0.739700373388$$

$$96 \times 1.34784891533 = 129.393495872$$ while $$96 \times 0.739700373388 = 71.0112358453$$

Answer: Between 71 and 130 inches.